Solving ../../benchmarks/true/max_nat.smt2...

Inference procedure has parameters:
Ice fuel: 200
Timeout: 30s
Convolution: complete


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(leq, P:
{() -> leq([z, n2])
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)])
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2])
(leq([s(nn1), z])) -> BOT}
)
(max, F:
{(leq([n, m])) -> max([n, m, m])
(not leq([n, m])) -> max([n, m, n])}
(max([_gp, _hp, _ip]) /\ max([_gp, _hp, _jp])) -> eq_nat([_ip, _jp])
)
}

properties:
{(max([i, j, _kp])) -> leq([i, _kp]), (max([i, j, _lp])) -> leq([j, _lp])}


over-approximation: {max}
under-approximation: {leq}

Clause system for inference is:

{
() -> leq([z, n2]) -> 0  ;  (leq([n, m])) -> max([n, m, m]) -> 0  ;  (not leq([n, m])) -> max([n, m, n]) -> 0  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 0  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 0  ;  (leq([s(nn1), z])) -> BOT -> 0  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 0  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 0
}


Solving took 0.095206 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051, q_gen_4054},
Q_f={q_gen_4047},
Delta=
{
<s>(q_gen_4051) -> q_gen_4051
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4054
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4052, q_gen_4053, q_gen_4058},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<s, s>(q_gen_4058) -> q_gen_4058
<z, z>() -> q_gen_4058
<s, s, s>(q_gen_4048) -> q_gen_4048
<s, z, s>(q_gen_4058) -> q_gen_4048
<z, s, s>(q_gen_4058) -> q_gen_4048
<z, z, z>() -> q_gen_4048
<z, s, z>(q_gen_4053) -> q_gen_4052
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.004135 s (model generation: 0.003935,  model checking: 0.000200):

Model:
|_
{
leq -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 0  ;  (leq([n, m])) -> max([n, m, m]) -> 0  ;  (not leq([n, m])) -> max([n, m, n]) -> 3  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1  ;  (leq([s(nn1), z])) -> BOT -> 1  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 1  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 1
}
Sat witness: Yes: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> z
})

-------------------------------------------
Step 1, which took 0.003620 s (model generation: 0.003555,  model checking: 0.000065):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047},
Q_f={},
Delta=
{
<z, z>() -> q_gen_4047
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048},
Q_f={q_gen_4048},
Delta=
{
<z, z, z>() -> q_gen_4048
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3  ;  (leq([n, m])) -> max([n, m, m]) -> 1  ;  (not leq([n, m])) -> max([n, m, n]) -> 3  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1  ;  (leq([s(nn1), z])) -> BOT -> 1  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 1  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 1
}
Sat witness: Yes: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 2, which took 0.003459 s (model generation: 0.003353,  model checking: 0.000106):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047},
Q_f={q_gen_4047},
Delta=
{
<z, z>() -> q_gen_4047
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048},
Q_f={},
Delta=
{
<z, z, z>() -> q_gen_4048
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3  ;  (leq([n, m])) -> max([n, m, m]) -> 1  ;  (not leq([n, m])) -> max([n, m, n]) -> 3  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 2  ;  (leq([s(nn1), z])) -> BOT -> 2  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 2  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 2
}
Sat witness: Yes: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 3, which took 0.003439 s (model generation: 0.003383,  model checking: 0.000056):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047},
Q_f={q_gen_4047},
Delta=
{
<s, s>(q_gen_4047) -> q_gen_4047
<z, z>() -> q_gen_4047
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048},
Q_f={},
Delta=
{
<z, z, z>() -> q_gen_4048
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3  ;  (leq([n, m])) -> max([n, m, m]) -> 4  ;  (not leq([n, m])) -> max([n, m, n]) -> 3  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 2  ;  (leq([s(nn1), z])) -> BOT -> 2  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 2  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 2
}
Sat witness: Yes: ((leq([n, m])) -> max([n, m, m]), {
m -> z  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.003793 s (model generation: 0.003348,  model checking: 0.000445):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047},
Q_f={q_gen_4047},
Delta=
{
<s, s>(q_gen_4047) -> q_gen_4047
<z, z>() -> q_gen_4047
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048},
Q_f={q_gen_4048},
Delta=
{
<z, z, z>() -> q_gen_4048
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3  ;  (leq([n, m])) -> max([n, m, m]) -> 4  ;  (not leq([n, m])) -> max([n, m, n]) -> 6  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 3  ;  (leq([s(nn1), z])) -> BOT -> 3  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 3  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 3
}
Sat witness: Yes: ((not leq([n, m])) -> max([n, m, n]), {
m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 5, which took 0.005011 s (model generation: 0.004929,  model checking: 0.000082):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051},
Q_f={q_gen_4047},
Delta=
{
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4053},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, s, z>(q_gen_4053) -> q_gen_4048
<z, z, z>() -> q_gen_4048
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3  ;  (leq([n, m])) -> max([n, m, m]) -> 4  ;  (not leq([n, m])) -> max([n, m, n]) -> 6  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 3  ;  (leq([s(nn1), z])) -> BOT -> 3  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 3  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Yes: ((max([i, j, _lp])) -> leq([j, _lp]), {
_lp -> z  ;  i -> z  ;  j -> s(z)
})

-------------------------------------------
Step 6, which took 0.007929 s (model generation: 0.007805,  model checking: 0.000124):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051},
Q_f={q_gen_4047},
Delta=
{
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4053},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, s, z>(q_gen_4053) -> q_gen_4048
<z, z, z>() -> q_gen_4048
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3  ;  (leq([n, m])) -> max([n, m, m]) -> 4  ;  (not leq([n, m])) -> max([n, m, n]) -> 6  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 3  ;  (leq([s(nn1), z])) -> BOT -> 6  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 4  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Yes: ((leq([s(nn1), z])) -> BOT, {
nn1 -> z
})

-------------------------------------------
Step 7, which took 0.010894 s (model generation: 0.010758,  model checking: 0.000136):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051, q_gen_4054},
Q_f={q_gen_4047},
Delta=
{
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4054
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4052, q_gen_4053},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, z, z>() -> q_gen_4048
<z, s, z>(q_gen_4053) -> q_gen_4052
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6  ;  (leq([n, m])) -> max([n, m, m]) -> 4  ;  (not leq([n, m])) -> max([n, m, n]) -> 6  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 4  ;  (leq([s(nn1), z])) -> BOT -> 6  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 4  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Yes: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

-------------------------------------------
Step 8, which took 0.008764 s (model generation: 0.008312,  model checking: 0.000452):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051, q_gen_4054},
Q_f={q_gen_4047},
Delta=
{
<s>(q_gen_4051) -> q_gen_4051
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4054
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4052, q_gen_4053},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, z, z>() -> q_gen_4048
<z, s, z>(q_gen_4053) -> q_gen_4052
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6  ;  (leq([n, m])) -> max([n, m, m]) -> 7  ;  (not leq([n, m])) -> max([n, m, n]) -> 6  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 5  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 5  ;  (leq([s(nn1), z])) -> BOT -> 6  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 5  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Yes: ((leq([n, m])) -> max([n, m, m]), {
m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.008219 s (model generation: 0.007875,  model checking: 0.000344):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051, q_gen_4054},
Q_f={q_gen_4047},
Delta=
{
<s>(q_gen_4051) -> q_gen_4051
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4054
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4052, q_gen_4053, q_gen_4058},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, z>() -> q_gen_4058
<z, s, s>(q_gen_4058) -> q_gen_4048
<z, z, z>() -> q_gen_4048
<z, s, z>(q_gen_4053) -> q_gen_4052
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6  ;  (leq([n, m])) -> max([n, m, m]) -> 7  ;  (not leq([n, m])) -> max([n, m, n]) -> 9  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 6  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 6  ;  (leq([s(nn1), z])) -> BOT -> 7  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 6  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 7
}
Sat witness: Yes: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> s(z)
})

-------------------------------------------
Step 10, which took 0.006292 s (model generation: 0.005073,  model checking: 0.001219):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051, q_gen_4054},
Q_f={q_gen_4047},
Delta=
{
<s>(q_gen_4051) -> q_gen_4051
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4054
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4052, q_gen_4053, q_gen_4058},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, z>() -> q_gen_4058
<s, z, s>(q_gen_4058) -> q_gen_4048
<z, s, s>(q_gen_4058) -> q_gen_4048
<z, z, z>() -> q_gen_4048
<z, s, z>(q_gen_4053) -> q_gen_4052
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 7  ;  (leq([n, m])) -> max([n, m, m]) -> 10  ;  (not leq([n, m])) -> max([n, m, n]) -> 9  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 7  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 7  ;  (leq([s(nn1), z])) -> BOT -> 8  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 7  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 8
}
Sat witness: Yes: ((leq([n, m])) -> max([n, m, m]), {
m -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 11, which took 0.014038 s (model generation: 0.012917,  model checking: 0.001121):

Model:
|_
{
leq -> 
{{{
Q={q_gen_4047, q_gen_4051, q_gen_4054},
Q_f={q_gen_4047},
Delta=
{
<s>(q_gen_4051) -> q_gen_4051
<z>() -> q_gen_4051
<s, s>(q_gen_4047) -> q_gen_4047
<z, s>(q_gen_4051) -> q_gen_4047
<z, z>() -> q_gen_4047
<s, z>(q_gen_4051) -> q_gen_4054
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  max -> 
{{{
Q={q_gen_4048, q_gen_4052, q_gen_4053, q_gen_4058},
Q_f={q_gen_4048},
Delta=
{
<z>() -> q_gen_4053
<z, z>() -> q_gen_4058
<s, s, s>(q_gen_4048) -> q_gen_4048
<s, z, s>(q_gen_4058) -> q_gen_4048
<z, s, s>(q_gen_4058) -> q_gen_4048
<z, z, z>() -> q_gen_4048
<z, s, z>(q_gen_4053) -> q_gen_4052
}

Datatype: <nat, nat, nat>
Convolution form: complete
}}}

}
--
Equality automata are defined for: {eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 8  ;  (leq([n, m])) -> max([n, m, m]) -> 10  ;  (not leq([n, m])) -> max([n, m, n]) -> 12  ;  (leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 8  ;  (leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 8  ;  (leq([s(nn1), z])) -> BOT -> 9  ;  (max([i, j, _kp])) -> leq([i, _kp]) -> 8  ;  (max([i, j, _lp])) -> leq([j, _lp]) -> 9
}
Sat witness: Yes: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> s(s(z))
})

Total time: 0.095206
Reason for stopping: Proved