Solving ../../benchmarks/smtlib/false/tree_depth_subtree_error.smt2...

Inference procedure has parameters:
Ice fuel: 200
Timeout: Some(60.) (sec)
Teacher_type: Checks all clauses every time
Approximation method: remove every clause that can be safely removed



Learning problem is:

env: {
elt -> {a, b}  ;  etree -> {leaf, node}  ;  nat -> {s, z}
}
definition:
{
(leq_nat, P:
{
leq_nat(z, n2) <= True
leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2)
leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2))
False <= leq_nat(s(nn1), z)
}
)
(le_nat, P:
{
le_nat(z, s(nn2)) <= True
le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2)
le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2))
False <= le_nat(s(nn1), z)
False <= le_nat(z, z)
}
)
(subtree, P:
{
subtree(leaf, t) <= True
subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2)
subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))
False <= subtree(node(ea, ta1, ta2), leaf)
eq_elt(ea, eb) <= subtree(node(ea, ta1, ta2), node(eb, tb1, tb2))
subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))
}
)
(max, F:
{
le_nat(n, m) \/ max(n, m, n) <= True
max(n, m, m) <= le_nat(n, m)
}
eq_nat(_ggb, _hgb) <= max(_egb, _fgb, _ggb) /\ max(_egb, _fgb, _hgb)
)
(height, F:
{
height(leaf, z) <= True
height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb)
height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb)
}
eq_nat(_ngb, _ogb) <= height(_mgb, _ngb) /\ height(_mgb, _ogb)
)
}

properties:
{
le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2)
}


over-approximation: {height, leq_nat, max, subtree}
under-approximation: {leq_nat, max}

Clause system for inference is:

{
height(leaf, z) <= True -> 0
le_nat(z, s(nn2)) <= True -> 0
subtree(leaf, t) <= True -> 0
height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb) -> 0
height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb) -> 0
le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2) -> 0
le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2) -> 0
le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2)) -> 0
False <= le_nat(s(nn1), z) -> 0
False <= le_nat(z, z) -> 0
leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2) -> 0
leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2)) -> 0
subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2) -> 0
subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
}


Solving took 0.044516 seconds.
No: Contradictory set of ground constraints:
{
False <= True
height(leaf, z) <= True
height(node(a, leaf, leaf), s(z)) <= True
le_nat(s(z), s(s(z))) <= True
le_nat(z, s(z)) <= True
subtree(leaf, leaf) <= True
subtree(leaf, node(a, leaf, leaf)) <= True
subtree(node(a, leaf, leaf), node(a, node(a, leaf, leaf), node(a, leaf, leaf))) <= True
False <= le_nat(s(s(z)), s(z))
False <= le_nat(s(z), s(z))
False <= le_nat(s(z), z)
False <= le_nat(z, z)
subtree(node(a, leaf, leaf), leaf) <= subtree(node(a, leaf, leaf), node(a, leaf, leaf)) /\ subtree(node(a, node(a, leaf, leaf), node(a, leaf, leaf)), node(a, node(a, leaf, leaf), leaf))
subtree(node(a, leaf, leaf), leaf) <= subtree(node(a, node(a, leaf, leaf), leaf), node(a, leaf, leaf))
}
-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.005907 s (model generation: 0.005802,  model checking: 0.000105):

Clauses:
{
height(leaf, z) <= True -> 0
le_nat(z, s(nn2)) <= True -> 0
subtree(leaf, t) <= True -> 0
height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb) -> 0
height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb) -> 0
le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2) -> 0
le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2) -> 0
le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2)) -> 0
False <= le_nat(s(nn1), z) -> 0
False <= le_nat(z, z) -> 0
leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2) -> 0
leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2)) -> 0
subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2) -> 0
subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
}

Accumulated learning constraints:
{

}

Current best model:
|_
name: None

height ->
[
height : <etree * nat>
{
  
}
]


;
le_nat ->
[
le_nat : <nat * nat>
{
  
}
]


;
leq_nat ->
[
leq_nat : <nat * nat>
{
  
}
]


;
subtree ->
[
subtree : <etree * etree>
{
  
}
]



--
Equality automata are defined for: {elt, etree, nat}
_|




Answer of teacher:

height(leaf, z) <= True  :
Yes: {

}

le_nat(z, s(nn2)) <= True  :
Yes: {

}

subtree(leaf, t) <= True  :
Yes: {
t -> node(_vmtqw_0, _vmtqw_1, _vmtqw_2)
}

height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb)  :
No: ()

height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb)  :
No: ()

le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2)  :
No: ()

le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2)  :
No: ()

le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2))  :
No: ()

False <= le_nat(s(nn1), z)  :
No: ()

False <= le_nat(z, z)  :
No: ()

leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2)  :
No: ()

leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2))  :
No: ()

subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2)  :
No: ()

subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
No: ()

subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
No: ()


-------------------------------------------
Step 1, which took 0.006957 s (model generation: 0.006832,  model checking: 0.000125):

Clauses:
{
height(leaf, z) <= True -> 0
le_nat(z, s(nn2)) <= True -> 0
subtree(leaf, t) <= True -> 0
height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb) -> 0
height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb) -> 0
le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2) -> 0
le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2) -> 0
le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2)) -> 0
False <= le_nat(s(nn1), z) -> 0
False <= le_nat(z, z) -> 0
leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2) -> 0
leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2)) -> 0
subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2) -> 0
subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
}

Accumulated learning constraints:
{
height(leaf, z) <= True
le_nat(z, s(z)) <= True
subtree(leaf, node(a, leaf, leaf)) <= True
}

Current best model:
|_
name: None

height ->
[
height : <etree * nat>
{
  height(leaf, z) <= True
}
]


;
le_nat ->
[
le_nat : <nat * nat>
{
  le_nat(z, s(x_1_0)) <= True
}
]


;
leq_nat ->
[
leq_nat : <nat * nat>
{
  
}
]


;
subtree ->
[
subtree : <etree * etree>
{
  subtree(leaf, node(x_1_0, x_1_1, x_1_2)) <= True
}
]



--
Equality automata are defined for: {elt, etree, nat}
_|




Answer of teacher:

height(leaf, z) <= True  :
No: ()

le_nat(z, s(nn2)) <= True  :
No: ()

subtree(leaf, t) <= True  :
Yes: {
t -> leaf
}

height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb)  :
Yes: {
_igb -> z  ;  _jgb -> z  ;  _lgb -> z  ;  t1 -> leaf  ;  t2 -> leaf
}

height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb)  :
No: ()

le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2)  :
No: ()

le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2)  :
Yes: {
nn1 -> z  ;  nn2 -> s(_dntqw_0)
}

le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2))  :
No: ()

False <= le_nat(s(nn1), z)  :
No: ()

False <= le_nat(z, z)  :
No: ()

leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2)  :
No: ()

leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2))  :
No: ()

subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2)  :
Yes: {
ta1 -> leaf  ;  ta2 -> leaf  ;  tb1 -> node(_gntqw_0, _gntqw_1, _gntqw_2)  ;  tb2 -> node(_hntqw_0, _hntqw_1, _hntqw_2)
}

subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
No: ()

subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
No: ()


-------------------------------------------
Step 2, which took 0.009568 s (model generation: 0.009364,  model checking: 0.000204):

Clauses:
{
height(leaf, z) <= True -> 0
le_nat(z, s(nn2)) <= True -> 0
subtree(leaf, t) <= True -> 0
height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb) -> 0
height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb) -> 0
le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2) -> 0
le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2) -> 0
le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2)) -> 0
False <= le_nat(s(nn1), z) -> 0
False <= le_nat(z, z) -> 0
leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2) -> 0
leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2)) -> 0
subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2) -> 0
subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
}

Accumulated learning constraints:
{
height(leaf, z) <= True
height(node(a, leaf, leaf), s(z)) \/ le_nat(z, z) <= True
le_nat(s(z), s(s(z))) <= True
le_nat(z, s(z)) <= True
subtree(leaf, leaf) <= True
subtree(leaf, node(a, leaf, leaf)) <= True
subtree(node(a, leaf, leaf), node(a, node(a, leaf, leaf), node(a, leaf, leaf))) <= True
}

Current best model:
|_
name: None

height ->
[
height : <etree * nat>
{
  height(leaf, z) <= True
  height(node(x_0_0, x_0_1, x_0_2), s(x_1_0)) <= True
}
]


;
le_nat ->
[
le_nat : <nat * nat>
{
  le_nat(s(x_0_0), s(x_1_0)) <= True
  le_nat(z, s(x_1_0)) <= True
  le_nat(z, z) <= True
}
]


;
leq_nat ->
[
leq_nat : <nat * nat>
{
  
}
]


;
subtree ->
[
subtree : <etree * etree>
{
  subtree(leaf, leaf) <= True
  subtree(leaf, node(x_1_0, x_1_1, x_1_2)) <= True
  subtree(node(x_0_0, x_0_1, x_0_2), node(x_1_0, x_1_1, x_1_2)) <= True
}
]



--
Equality automata are defined for: {elt, etree, nat}
_|




Answer of teacher:

height(leaf, z) <= True  :
No: ()

le_nat(z, s(nn2)) <= True  :
No: ()

subtree(leaf, t) <= True  :
No: ()

height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb)  :
No: ()

height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb)  :
No: ()

le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2)  :
No: ()

le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2)  :
No: ()

le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2))  :
Yes: {
nn1 -> s(_intqw_0)  ;  nn2 -> z
}

False <= le_nat(s(nn1), z)  :
No: ()

False <= le_nat(z, z)  :
Yes: {

}

leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2)  :
No: ()

leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2))  :
No: ()

subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2)  :
No: ()

subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
Yes: {
ta1 -> node(_sntqw_0, _sntqw_1, _sntqw_2)  ;  ta2 -> node(_tntqw_0, _tntqw_1, _tntqw_2)  ;  tb1 -> node(_untqw_0, _untqw_1, _untqw_2)  ;  tb2 -> leaf
}

subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
Yes: {
ta1 -> node(_wntqw_0, _wntqw_1, _wntqw_2)  ;  tb1 -> leaf
}


-------------------------------------------
Step 3, which took 0.011474 s (model generation: 0.011333,  model checking: 0.000141):

Clauses:
{
height(leaf, z) <= True -> 0
le_nat(z, s(nn2)) <= True -> 0
subtree(leaf, t) <= True -> 0
height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb) -> 0
height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb) -> 0
le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2) -> 0
le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2) -> 0
le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2)) -> 0
False <= le_nat(s(nn1), z) -> 0
False <= le_nat(z, z) -> 0
leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2) -> 0
leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2)) -> 0
subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2) -> 0
subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) -> 0
}

Accumulated learning constraints:
{
height(leaf, z) <= True
height(node(a, leaf, leaf), s(z)) <= True
le_nat(s(z), s(s(z))) <= True
le_nat(z, s(z)) <= True
subtree(leaf, leaf) <= True
subtree(leaf, node(a, leaf, leaf)) <= True
subtree(node(a, leaf, leaf), node(a, node(a, leaf, leaf), node(a, leaf, leaf))) <= True
le_nat(s(z), z) <= le_nat(s(s(z)), s(z))
False <= le_nat(z, z)
subtree(node(a, leaf, leaf), leaf) <= subtree(node(a, leaf, leaf), node(a, leaf, leaf)) /\ subtree(node(a, node(a, leaf, leaf), node(a, leaf, leaf)), node(a, node(a, leaf, leaf), leaf))
subtree(node(a, leaf, leaf), leaf) <= subtree(node(a, node(a, leaf, leaf), leaf), node(a, leaf, leaf))
}

Current best model:
|_
name: None

height ->
[
height : <etree * nat>
{
  height(leaf, z) <= True
  height(node(x_0_0, x_0_1, x_0_2), s(x_1_0)) <= True
}
]


;
le_nat ->
[
le_nat : <nat * nat>
{
  le_nat(s(x_0_0), s(x_1_0)) <= True
  le_nat(s(x_0_0), z) <= True
  le_nat(z, s(x_1_0)) <= True
}
]


;
leq_nat ->
[
leq_nat : <nat * nat>
{
  
}
]


;
subtree ->
[
subtree : <etree * etree>
{
  subtree(leaf, leaf) <= True
  subtree(leaf, node(x_1_0, x_1_1, x_1_2)) <= True
  subtree(node(x_0_0, x_0_1, x_0_2), leaf) <= True
  subtree(node(x_0_0, x_0_1, x_0_2), node(x_1_0, x_1_1, x_1_2)) <= True
}
]



--
Equality automata are defined for: {elt, etree, nat}
_|




Answer of teacher:

height(leaf, z) <= True  :
No: ()

le_nat(z, s(nn2)) <= True  :
No: ()

subtree(leaf, t) <= True  :
No: ()

height(node(e, t1, t2), s(_lgb)) \/ le_nat(_igb, _jgb) <= height(t1, _igb) /\ height(t1, _lgb) /\ height(t2, _jgb)  :
No: ()

height(node(e, t1, t2), s(_kgb)) <= height(t1, _igb) /\ height(t2, _jgb) /\ height(t2, _kgb) /\ le_nat(_igb, _jgb)  :
No: ()

le_nat(_pgb, _qgb) <= height(t1, _pgb) /\ height(t2, _qgb) /\ subtree(t1, t2)  :
Yes: {
_pgb -> z  ;  _qgb -> z  ;  t1 -> leaf  ;  t2 -> leaf
}

le_nat(s(nn1), s(nn2)) <= le_nat(nn1, nn2)  :
No: ()

le_nat(nn1, nn2) <= le_nat(s(nn1), s(nn2))  :
Yes: {
nn1 -> z  ;  nn2 -> z
}

False <= le_nat(s(nn1), z)  :
Yes: {

}

False <= le_nat(z, z)  :
No: ()

leq_nat(s(nn1), s(nn2)) <= leq_nat(nn1, nn2)  :
No: ()

leq_nat(nn1, nn2) <= leq_nat(s(nn1), s(nn2))  :
No: ()

subtree(node(eb, ta1, ta2), node(eb, tb1, tb2)) <= subtree(ta1, tb1) /\ subtree(ta2, tb2)  :
No: ()

subtree(ta2, tb2) <= subtree(ta1, tb1) /\ subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
No: ()

subtree(ta1, tb1) <= subtree(node(eb, ta1, ta2), node(eb, tb1, tb2))  :
No: ()


Total time: 0.044516
Learner time: 0.033331
Teacher time: 0.000575
Reasons for stopping: No: Contradictory set of ground constraints:
{
False <= True
height(leaf, z) <= True
height(node(a, leaf, leaf), s(z)) <= True
le_nat(s(z), s(s(z))) <= True
le_nat(z, s(z)) <= True
subtree(leaf, leaf) <= True
subtree(leaf, node(a, leaf, leaf)) <= True
subtree(node(a, leaf, leaf), node(a, node(a, leaf, leaf), node(a, leaf, leaf))) <= True
False <= le_nat(s(s(z)), s(z))
False <= le_nat(s(z), s(z))
False <= le_nat(s(z), z)
False <= le_nat(z, z)
subtree(node(a, leaf, leaf), leaf) <= subtree(node(a, leaf, leaf), node(a, leaf, leaf)) /\ subtree(node(a, node(a, leaf, leaf), node(a, leaf, leaf)), node(a, node(a, leaf, leaf), leaf))
subtree(node(a, leaf, leaf), leaf) <= subtree(node(a, node(a, leaf, leaf), leaf), node(a, leaf, leaf))
}