┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "CD8 T cell differentiation"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: N, E, S, M, P
[ Info: Parameters: rho_P, mu_EE, delta_NE, mu_LE, delta_EL, mu_N, mu_M, delta_LM, mu_PE, mu_PL, mu_LL, mu_P, rho_E
[ Info: Inputs: 
[ Info: Outputs: y1, y2, y3
[ 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 15.362481139 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 15.362481139
[ Info: Computing Wronskians
┌ Info: Computed in 11.573056905 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.573056905
[ Info: Dimensions of the Wronskians [8, 14, 10]
┌ Info: Ranks of the Wronskians computed in 0.034015456 seconds
│   :rank_time = :rank_time
└   rank_times = 0.034015456

⌜ # 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: 1 for num. and 2 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 13.074297936 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.222819565 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 14 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 12 fractions 12 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 4.814101729 seconds. Result: true
[ Info: Out of 30 initial generators there are 12 indepdendent
[ Info: The ranking of the new set of generators is 159
[ 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: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 5.138238638 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.465960933 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 19 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (18 in total): Nemo.QQMPolyRingElem[rho_P, mu_EE, delta_NE, mu_LE, delta_EL, mu_N, mu_M, delta_LM, mu_PE, mu_PL, mu_LL, mu_P, rho_E, N, E, S, M, P]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 46
┌ Info: Final cleaning and simplification of generators. 
└ Out of 34 fractions 19 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.266522624 seconds. Result: true
[ Info: Out of 20 initial generators there are 17 indepdendent
[ Info: The ranking of the new set of generators is 159
[ Info: The search for identifiable functions concluded in 79.747776241 seconds
[ Info: Processing CD8 T cell differentiation
┌ 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 0.028282885 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.028282885
[ Info: Computing Wronskians
┌ Info: Computed in 0.022808084 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.022808084
[ Info: Dimensions of the Wronskians [8, 14, 10]
┌ Info: Ranks of the Wronskians computed in 4.3867e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 4.3867e-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: 1 for num. and 2 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 0.018989265 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003466853 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 14 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 12 fractions 12 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.003787761 seconds. Result: true
[ Info: Out of 30 initial generators there are 12 indepdendent
[ Info: The ranking of the new set of generators is 159
[ 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: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 0.020896246 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003529679 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 19 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (18 in total): Nemo.QQMPolyRingElem[rho_P, mu_EE, delta_NE, mu_LE, delta_EL, mu_N, mu_M, delta_LM, mu_PE, mu_PL, mu_LL, mu_P, rho_E, N, E, S, M, P]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 46
┌ Info: Final cleaning and simplification of generators. 
└ Out of 34 fractions 19 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.005065786 seconds. Result: true
[ Info: Out of 20 initial generators there are 17 indepdendent
[ Info: The ranking of the new set of generators is 159
[ Info: The search for identifiable functions concluded in 0.716978186 seconds
┌ Info: Result is
│   result =
│    17-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     M
│     S
│     E
│     N
│     ⋮
│     P*rho_E
│     delta_NE*P
└     rho_P*P
