Term Rewriting System R:
[x, y, z]
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)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(O(x), O(y)) -> O'(+(x, y))
+'(O(x), O(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)
+'(I(x), O(y)) -> +'(x, y)
+'(I(x), I(y)) -> O'(+(+(x, y), I(0)))
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), I(y)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)
-'(O(x), O(y)) -> O'(-(x, y))
-'(O(x), O(y)) -> -'(x, y)
-'(O(x), I(y)) -> -'(-(x, y), I(1))
-'(O(x), I(y)) -> -'(x, y)
-'(I(x), O(y)) -> -'(x, y)
-'(I(x), I(y)) -> O'(-(x, y))
-'(I(x), I(y)) -> -'(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)) -> +'(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)) -> +'(Log'(x), I(0))
LOG'(O(x)) -> LOG'(x)
LOG(x) -> -'(Log'(x), I(0))
LOG(x) -> LOG'(x)
MIN(N(x, l, r)) -> MIN(l)
MAX(N(x, l, r)) -> MAX(r)
BS'(N(x, l, r)) -> AND(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
BS'(N(x, l, r)) -> AND(ge(x, Max(l)), ge(Min(r), x))
BS'(N(x, l, r)) -> GE(x, Max(l))
BS'(N(x, l, r)) -> MAX(l)
BS'(N(x, l, r)) -> GE(Min(r), x)
BS'(N(x, l, r)) -> MIN(r)
BS'(N(x, l, r)) -> AND(BS(l), BS(r))
BS'(N(x, l, r)) -> BS'(l)
BS'(N(x, l, r)) -> BS'(r)
SIZE(N(x, l, r)) -> +'(+(Size(l), Size(r)), I(1))
SIZE(N(x, l, r)) -> +'(Size(l), Size(r))
SIZE(N(x, l, r)) -> SIZE(l)
SIZE(N(x, l, r)) -> SIZE(r)
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)))
WB'(N(x, l, r)) -> IF(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l))))
WB'(N(x, l, r)) -> GE(Size(l), Size(r))
WB'(N(x, l, r)) -> SIZE(l)
WB'(N(x, l, r)) -> SIZE(r)
WB'(N(x, l, r)) -> GE(I(0), -(Size(l), Size(r)))
WB'(N(x, l, r)) -> -'(Size(l), Size(r))
WB'(N(x, l, r)) -> GE(I(0), -(Size(r), Size(l)))
WB'(N(x, l, r)) -> -'(Size(r), Size(l))
WB'(N(x, l, r)) -> AND(WB(l), WB(r))
WB'(N(x, l, r)) -> WB'(l)
WB'(N(x, l, r)) -> WB'(r)

Furthermore, R contains five SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pairs:

+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), O(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)
+'(O(x), O(y)) -> +'(x, y)

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)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

+'(I(x), O(y)) -> +'(x, y)
+'(O(x), O(y)) -> +'(x, y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  x1 POL(0) =  0 POL(O(x1)) =  1 + x1 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 6`
`             ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pairs:

+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(O(x), I(y)) -> +'(x, y)

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)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  x1 POL(0) =  0 POL(O(x1)) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 6`
`             ↳Polo`
`             ...`
`               →DP Problem 7`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pairs:

+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(O(x), I(y)) -> +'(x, y)

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)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

+'(I(x), I(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  1 + x1 POL(0) =  0 POL(O(x1)) =  0 POL(+(x1, x2)) =  0 POL(+'(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 6`
`             ↳Polo`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

+'(I(x), I(y)) -> +'(+(x, y), I(0))

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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

+'(I(x), I(y)) -> +'(+(x, y), I(0))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

+(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)
O(0) -> 0

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  1 + x1 POL(0) =  0 POL(O(x1)) =  x1 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 6`
`             ↳Polo`
`             ...`
`               →DP Problem 9`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

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)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pairs:

-'(I(x), I(y)) -> -'(x, y)
-'(I(x), O(y)) -> -'(x, y)
-'(O(x), I(y)) -> -'(x, y)
-'(O(x), I(y)) -> -'(-(x, y), I(1))
-'(O(x), O(y)) -> -'(x, y)

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)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

-'(I(x), O(y)) -> -'(x, y)
-'(O(x), O(y)) -> -'(x, y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  x1 POL(0) =  0 POL(-'(x1, x2)) =  x2 POL(1) =  0 POL(O(x1)) =  1 + x1 POL(-(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 10`
`             ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pairs:

-'(I(x), I(y)) -> -'(x, y)
-'(O(x), I(y)) -> -'(x, y)
-'(O(x), I(y)) -> -'(-(x, y), I(1))

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)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

-'(I(x), I(y)) -> -'(x, y)
-'(O(x), I(y)) -> -'(x, y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  1 + x1 POL(0) =  0 POL(-'(x1, x2)) =  x2 POL(1) =  0 POL(O(x1)) =  0 POL(-(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 10`
`             ↳Polo`
`             ...`
`               →DP Problem 11`
`                 ↳Narrowing Transformation`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

-'(O(x), I(y)) -> -'(-(x, y), I(1))

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)))

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

-'(O(x), I(y)) -> -'(-(x, y), I(1))
six new Dependency Pairs are created:

-'(O(x''), I(0)) -> -'(x'', I(1))
-'(O(0), I(y')) -> -'(0, I(1))
-'(O(O(x'')), I(O(y''))) -> -'(O(-(x'', y'')), I(1))
-'(O(O(x'')), I(I(y''))) -> -'(I(-(-(x'', y''), I(1))), I(1))
-'(O(I(x'')), I(O(y''))) -> -'(I(-(x'', y'')), I(1))
-'(O(I(x'')), I(I(y''))) -> -'(O(-(x'', y'')), I(1))

The transformation is resulting in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

GE(0, O(x)) -> GE(0, x)

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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

GE(0, O(x)) -> GE(0, x)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(GE(x1, x2)) =  x2 POL(O(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 12`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

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)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pairs:

GE(I(x), I(y)) -> GE(x, y)
GE(I(x), O(y)) -> GE(x, y)
GE(O(x), I(y)) -> GE(y, x)
GE(O(x), O(y)) -> GE(x, y)

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)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

GE(I(x), I(y)) -> GE(x, y)
GE(I(x), O(y)) -> GE(x, y)
GE(O(x), I(y)) -> GE(y, x)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  1 + x1 POL(GE(x1, x2)) =  1 + x1 + x2 POL(O(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 13`
`             ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

GE(O(x), O(y)) -> GE(x, y)

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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

GE(O(x), O(y)) -> GE(x, y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(GE(x1, x2)) =  x1 POL(O(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 13`
`             ↳Polo`
`             ...`
`               →DP Problem 14`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Polo`

Dependency Pair:

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)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polynomial Ordering`

Dependency Pairs:

LOG'(O(x)) -> LOG'(x)
LOG'(I(x)) -> LOG'(x)

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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

LOG'(O(x)) -> LOG'(x)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(LOG'(x1)) =  x1 POL(I(x1)) =  x1 POL(O(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`           →DP Problem 15`
`             ↳Polynomial Ordering`

Dependency Pair:

LOG'(I(x)) -> LOG'(x)

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)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

LOG'(I(x)) -> LOG'(x)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(LOG'(x1)) =  x1 POL(I(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`           →DP Problem 15`
`             ↳Polo`
`             ...`
`               →DP Problem 16`
`                 ↳Dependency Graph`

Dependency Pair:

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)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes