Inference procedure has parameters:
Ice fuel: 200
Timeout: 60s
Convolution: right


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(plus, F:
{() -> plus([n, z, n])
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)])}
(plus([_bp, _cp, _dp]) /\ plus([_bp, _cp, _ep])) -> eq_nat([_dp, _ep])
)
(leq, P:
{() -> leq([z, s(nn2)])
() -> leq([z, z])
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)])
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2])
(leq([s(nn1), z])) -> BOT}
)
}

properties:
{(plus([n, m, _fp])) -> leq([n, _fp])}


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

Clause system for inference is:

{
() -> leq([z, s(nn2)]) -> 0
() -> leq([z, z]) -> 0
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 0
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 0
}


Solving took 0.136768 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<s>(q_gen_3040) -> q_gen_3040
<z>() -> q_gen_3040
<s, s>(q_gen_3038) -> q_gen_3038
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042, q_gen_3052},
Q_f={q_gen_3037},
Delta=
{
<s>(q_gen_3052) -> q_gen_3052
<z>() -> q_gen_3052
<s, s>(q_gen_3042) -> q_gen_3042
<z, s>(q_gen_3052) -> q_gen_3042
<z, z>() -> q_gen_3042
<s, s, s>(q_gen_3037) -> q_gen_3037
<s, z, s>(q_gen_3042) -> q_gen_3037
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, s>(q_gen_3052) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010228 s (model generation: 0.009855,  model checking: 0.000373):

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

}

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

}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 0
() -> leq([z, z]) -> 0
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 1
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.010210 s (model generation: 0.010124,  model checking: 0.000086):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037},
Q_f={q_gen_3037},
Delta=
{
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 0
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 1
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 1
}
Sat witness: Found: (() -> leq([z, z]), {

})

-------------------------------------------
Step 2, which took 0.010299 s (model generation: 0.010175,  model checking: 0.000124):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038},
Q_f={q_gen_3038},
Delta=
{
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037},
Q_f={q_gen_3037},
Delta=
{
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 1
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 1
}
Sat witness: Found: (() -> leq([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 3, which took 0.010664 s (model generation: 0.010472,  model checking: 0.000192):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<z>() -> q_gen_3040
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037},
Q_f={q_gen_3037},
Delta=
{
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 1
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]), {
_ap -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.011160 s (model generation: 0.010828,  model checking: 0.000332):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<z>() -> q_gen_3040
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042},
Q_f={q_gen_3037},
Delta=
{
<z, z>() -> q_gen_3042
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 2
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 5, which took 0.012477 s (model generation: 0.011989,  model checking: 0.000488):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<z>() -> q_gen_3040
<s, s>(q_gen_3038) -> q_gen_3038
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042},
Q_f={q_gen_3037},
Delta=
{
<z, z>() -> q_gen_3042
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 3
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.011604 s (model generation: 0.011267,  model checking: 0.000337):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<z>() -> q_gen_3040
<s, s>(q_gen_3038) -> q_gen_3038
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042},
Q_f={q_gen_3037},
Delta=
{
<s, s>(q_gen_3042) -> q_gen_3042
<z, z>() -> q_gen_3042
<s, z, s>(q_gen_3042) -> q_gen_3037
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _fp])) -> leq([n, _fp]) -> 6
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 4
}
Sat witness: Found: ((plus([n, m, _fp])) -> leq([n, _fp]), {
_fp -> s(s(z))  ;  m -> s(s(z))  ;  n -> z
})

-------------------------------------------
Step 7, which took 0.013195 s (model generation: 0.012218,  model checking: 0.000977):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<s>(q_gen_3040) -> q_gen_3040
<z>() -> q_gen_3040
<s, s>(q_gen_3038) -> q_gen_3038
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042},
Q_f={q_gen_3037},
Delta=
{
<s, s>(q_gen_3042) -> q_gen_3042
<z, z>() -> q_gen_3042
<s, z, s>(q_gen_3042) -> q_gen_3037
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 4
() -> leq([z, z]) -> 4
() -> plus([n, z, 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 -> 4
(plus([n, m, _fp])) -> leq([n, _fp]) -> 6
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]), {
_ap -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.014211 s (model generation: 0.013233,  model checking: 0.000978):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<s>(q_gen_3040) -> q_gen_3040
<z>() -> q_gen_3040
<s, s>(q_gen_3038) -> q_gen_3038
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042, q_gen_3052},
Q_f={q_gen_3037},
Delta=
{
<z>() -> q_gen_3052
<s, s>(q_gen_3042) -> q_gen_3042
<z, z>() -> q_gen_3042
<s, s, s>(q_gen_3037) -> q_gen_3037
<s, z, s>(q_gen_3042) -> q_gen_3037
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, s>(q_gen_3052) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 5
() -> leq([z, z]) -> 5
() -> plus([n, z, n]) -> 7
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 5
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 5
(leq([s(nn1), z])) -> BOT -> 5
(plus([n, m, _fp])) -> leq([n, _fp]) -> 7
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]), {
_ap -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.015377 s (model generation: 0.013972,  model checking: 0.001405):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3038, q_gen_3040},
Q_f={q_gen_3038},
Delta=
{
<s>(q_gen_3040) -> q_gen_3040
<z>() -> q_gen_3040
<s, s>(q_gen_3038) -> q_gen_3038
<z, s>(q_gen_3040) -> q_gen_3038
<z, z>() -> q_gen_3038
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_3037, q_gen_3042, q_gen_3052},
Q_f={q_gen_3037},
Delta=
{
<z>() -> q_gen_3052
<s, s>(q_gen_3042) -> q_gen_3042
<z, s>(q_gen_3052) -> q_gen_3042
<z, z>() -> q_gen_3042
<s, s, s>(q_gen_3037) -> q_gen_3037
<s, z, s>(q_gen_3042) -> q_gen_3037
<z, s, s>(q_gen_3042) -> q_gen_3037
<z, z, s>(q_gen_3052) -> q_gen_3037
<z, z, z>() -> q_gen_3037
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 6
() -> leq([z, z]) -> 6
() -> plus([n, z, n]) -> 8
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 6
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 6
(leq([s(nn1), z])) -> BOT -> 6
(plus([n, m, _fp])) -> leq([n, _fp]) -> 8
(plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _ap])) -> plus([n, s(mm), s(_ap)]), {
_ap -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 0.136768
Reason for stopping: Proved