Termination w.r.t. Q proof of /tmp/tmpdrpH7i/14.trs

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

  ↳11 QDP

    ↳12 QDPOrderProof (⇔)

    ↳13 QDP

    ↳14 DependencyGraphProof (⇔)

    ↳15 AND

      ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 PisEmptyProof (⇔)

        ↳20 TRUE

      ↳21 QDP

        ↳22 QDPOrderProof (⇔)

        ↳23 QDP

        ↳24 QDPOrderProof (⇔)

        ↳25 QDP

        ↳26 PisEmptyProof (⇔)

        ↳27 TRUE

  ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 DependencyGraphProof (⇔)

    ↳32 AND

      ↳33 QDP

      ↳34 QDP

        ↳35 QDPOrderProof (⇔)

        ↳36 QDP

        ↳37 QDPOrderProof (⇔)

        ↳38 QDP

        ↳39 PisEmptyProof (⇔)

        ↳40 TRUE

  ↳41 QDP

    ↳42 QDPOrderProof (⇔)

    ↳43 QDP

    ↳44 QDPOrderProof (⇔)

    ↳45 QDP

    ↳46 PisEmptyProof (⇔)

    ↳47 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+¹(O(x), O(y)) → O¹(+(x, y))
+¹(O(x), O(y)) → +¹(x, y)
+¹(O(x), I(y)) → +¹(x, y)
+¹(I(x), O(y)) → +¹(x, y)
+¹(I(x), I(y)) → O¹(+(+(x, y), I(0)))
+¹(I(x), I(y)) → +¹(+(x, y), I(0))
+¹(I(x), I(y)) → +¹(x, y)
+¹(x, +(y, z)) → +¹(+(x, y), z)
+¹(x, +(y, z)) → +¹(x, y)
-¹(O(x), O(y)) → O¹(-(x, y))
-¹(O(x), O(y)) → -¹(x, y)
-¹(O(x), I(y)) → -¹(-(x, y), I(1))
-¹(O(x), I(y)) → -¹(x, y)
-¹(I(x), O(y)) → -¹(x, y)
-¹(I(x), I(y)) → O¹(-(x, y))
-¹(I(x), I(y)) → -¹(x, y)
GE(O(x), O(y)) → GE(x, y)
GE(O(x), I(y)) → NOT(ge(y, x))
GE(O(x), I(y)) → GE(y, x)
GE(I(x), O(y)) → GE(x, y)
GE(I(x), I(y)) → GE(x, y)
GE(0, O(x)) → GE(0, x)
LOG'(I(x)) → +¹(Log'(x), I(0))
LOG'(I(x)) → LOG'(x)
LOG'(O(x)) → IF(ge(x, I(0)), +(Log'(x), I(0)), 0)
LOG'(O(x)) → GE(x, I(0))
LOG'(O(x)) → +¹(Log'(x), I(0))
LOG'(O(x)) → LOG'(x)
LOG(x) → -¹(Log'(x), I(0))
LOG(x) → LOG'(x)
MIN(N(x, l, r)) → MIN(l)
MAX(N(x, l, r)) → MAX(r)
BS¹(N(x, l, r)) → AND(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
BS¹(N(x, l, r)) → AND(ge(x, Max(l)), ge(Min(r), x))
BS¹(N(x, l, r)) → GE(x, Max(l))
BS¹(N(x, l, r)) → MAX(l)
BS¹(N(x, l, r)) → GE(Min(r), x)
BS¹(N(x, l, r)) → MIN(r)
BS¹(N(x, l, r)) → AND(BS(l), BS(r))
BS¹(N(x, l, r)) → BS¹(l)
BS¹(N(x, l, r)) → BS¹(r)
SIZE(N(x, l, r)) → +¹(+(Size(l), Size(r)), I(1))
SIZE(N(x, l, r)) → +¹(Size(l), Size(r))
SIZE(N(x, l, r)) → SIZE(l)
SIZE(N(x, l, r)) → SIZE(r)
WB¹(N(x, l, r)) → AND(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))
WB¹(N(x, l, r)) → IF(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l))))
WB¹(N(x, l, r)) → GE(Size(l), Size(r))
WB¹(N(x, l, r)) → SIZE(l)
WB¹(N(x, l, r)) → SIZE(r)
WB¹(N(x, l, r)) → GE(I(0), -(Size(l), Size(r)))
WB¹(N(x, l, r)) → -¹(Size(l), Size(r))
WB¹(N(x, l, r)) → GE(I(0), -(Size(r), Size(l)))
WB¹(N(x, l, r)) → -¹(Size(r), Size(l))
WB¹(N(x, l, r)) → AND(WB(l), WB(r))
WB¹(N(x, l, r)) → WB¹(l)
WB¹(N(x, l, r)) → WB¹(r)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 38 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GE(0, O(x)) → GE(0, x)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

GE(0, O(x)) → GE(0, x)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
GE(x1, x2) = x2
0 = 0
O(x1) = O(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

[0, O₁]

Status:

0: []
O₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GE(O(x), I(y)) → GE(y, x)
GE(O(x), O(y)) → GE(x, y)
GE(I(x), O(y)) → GE(x, y)
GE(I(x), I(y)) → GE(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

-¹(O(x), I(y)) → -¹(-(x, y), I(1))
-¹(O(x), I(y)) → -¹(x, y)
-¹(O(x), O(y)) → -¹(x, y)
-¹(I(x), O(y)) → -¹(x, y)
-¹(I(x), I(y)) → -¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

-¹(O(x), I(y)) → -¹(x, y)
-¹(I(x), I(y)) → -¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-¹(x1, x2) = x2
O(x1) = x1
I(x1) = I(x1)
-(x1, x2) = -
1 = 1
0 = 0

Lexicographic path order with status [LPO].
Quasi-Precedence:

- > 0 > [I₁, 1]

Status:

I₁: [1]
-: []
1: []
0: []

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

-¹(O(x), I(y)) → -¹(-(x, y), I(1))
-¹(O(x), O(y)) → -¹(x, y)
-¹(I(x), O(y)) → -¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(14) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(15) Complex Obligation (AND)

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

-¹(O(x), I(y)) → -¹(-(x, y), I(1))

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

-¹(O(x), I(y)) → -¹(-(x, y), I(1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-¹(x1, x2) = -¹(x1, x2)
O(x1) = O(x1)
I(x1) = I(x1)
-(x1, x2) = x1
1 = 1
0 = 0

Lexicographic path order with status [LPO].
Quasi-Precedence:

-^1₂ > [O₁, I₁] > 1
0 > 1

Status:

-^1₂: [2,1]
O₁: [1]
I₁: [1]
1: []
0: []

The following usable rules [FROCOS05] were oriented:

-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
O(0) → 0

(18) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) TRUE

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

-¹(I(x), O(y)) → -¹(x, y)
-¹(O(x), O(y)) → -¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

-¹(I(x), O(y)) → -¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-¹(x1, x2) = -¹(x1, x2)
I(x1) = I(x1)
O(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[-^1₂, I₁]

Status:

-^1₂: [2,1]
I₁: [1]

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

Q DP problem:
The TRS P consists of the following rules:

-¹(O(x), O(y)) → -¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

-¹(O(x), O(y)) → -¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-¹(x1, x2) = -¹(x2)
O(x1) = O(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

-^1₁: [1]
O₁: [1]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(26) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(27) TRUE

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+¹(O(x), I(y)) → +¹(x, y)
+¹(O(x), O(y)) → +¹(x, y)
+¹(I(x), O(y)) → +¹(x, y)
+¹(I(x), I(y)) → +¹(+(x, y), I(0))
+¹(I(x), I(y)) → +¹(x, y)
+¹(x, +(y, z)) → +¹(+(x, y), z)
+¹(x, +(y, z)) → +¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

+¹(O(x), I(y)) → +¹(x, y)
+¹(I(x), I(y)) → +¹(x, y)
+¹(x, +(y, z)) → +¹(+(x, y), z)
+¹(x, +(y, z)) → +¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
+¹(x1, x2) = +¹(x2)
O(x1) = x1
I(x1) = I(x1)
+(x1, x2) = +(x1, x2)
0 = 0

Lexicographic path order with status [LPO].
Quasi-Precedence:

+^1₁ > [I₁, +₂] > 0

Status:

+^1₁: [1]
I₁: [1]
+₂: [1,2]
0: []

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+¹(O(x), O(y)) → +¹(x, y)
+¹(I(x), O(y)) → +¹(x, y)
+¹(I(x), I(y)) → +¹(+(x, y), I(0))

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(32) Complex Obligation (AND)

(33) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+¹(I(x), I(y)) → +¹(+(x, y), I(0))

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+¹(I(x), O(y)) → +¹(x, y)
+¹(O(x), O(y)) → +¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

+¹(I(x), O(y)) → +¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
+¹(x1, x2) = +¹(x1, x2)
I(x1) = I(x1)
O(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[+^1₂, I₁]

Status:

+^1₂: [2,1]
I₁: [1]

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+¹(O(x), O(y)) → +¹(x, y)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

+¹(O(x), O(y)) → +¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
+¹(x1, x2) = +¹(x2)
O(x1) = O(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

+^1₁: [1]
O₁: [1]

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(40) TRUE

(41) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOG'(O(x)) → LOG'(x)
LOG'(I(x)) → LOG'(x)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LOG'(O(x)) → LOG'(x)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LOG'(x1) = LOG'(x1)
O(x1) = O(x1)
I(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[LOG'₁, O₁]

Status:

LOG'₁: [1]
O₁: [1]

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOG'(I(x)) → LOG'(x)

The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LOG'(I(x)) → LOG'(x)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LOG'(x1) = x1
I(x1) = I(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

I₁: [1]

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(46) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) Obligation:

(12) QDPOrderProof (EQUIVALENT transformation)

(13) Obligation:

(14) DependencyGraphProof (EQUIVALENT transformation)

(15) Complex Obligation (AND)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) PisEmptyProof (EQUIVALENT transformation)

(20) TRUE

(21) Obligation:

(22) QDPOrderProof (EQUIVALENT transformation)

(23) Obligation:

(24) QDPOrderProof (EQUIVALENT transformation)

(25) Obligation:

(26) PisEmptyProof (EQUIVALENT transformation)

(27) TRUE

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) DependencyGraphProof (EQUIVALENT transformation)

(32) Complex Obligation (AND)

(33) Obligation:

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) PisEmptyProof (EQUIVALENT transformation)

(40) TRUE

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) PisEmptyProof (EQUIVALENT transformation)

(47) TRUE