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`
`         ↳Usable Rules (Innermost)`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

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

As we are in the innermost case, we can delete all 35 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 6`
`             ↳Modular Removal of Rules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

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:

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

Strategy:

innermost

We have the following set of usable rules:

+(I(x), O(y)) -> I(+(x, y))
+(x, 0) -> x
+(0, x) -> x
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
+(O(x), O(y)) -> O(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
+(O(x), I(y)) -> I(+(x, y))
O(0) -> 0
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
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

We have the following set D of usable symbols: {I, 0, O, +, +'}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

+'(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)

The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

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

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 6`
`             ↳MRR`
`             ...`
`               →DP Problem 7`
`                 ↳Modular Removal of Rules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

We have the following set of usable rules:

+(I(x), O(y)) -> I(+(x, y))
+(x, 0) -> x
+(0, x) -> x
+(O(x), O(y)) -> O(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
O(0) -> 0
+(O(x), I(y)) -> I(+(x, y))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(I(x1)) =  x1 POL(0) =  0 POL(O(x1)) =  1 + x1 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  x1 + x2

We have the following set D of usable symbols: {I, 0, O, +, +'}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

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

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 6`
`             ↳MRR`
`             ...`
`               →DP Problem 8`
`                 ↳Modular Removal of Rules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

Dependency Pairs:

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

Rules:

+(x, 0) -> x
+(0, x) -> x
+(x, +(y, z)) -> +(+(x, y), z)

Strategy:

innermost

We have the following set of usable rules:

+(x, 0) -> x
+(0, x) -> x
+(x, +(y, z)) -> +(+(x, y), z)
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(0) =  0 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  1 + x1 + x2

We have the following set D of usable symbols: {+, +'}
No Dependency Pairs can be deleted.
The following rules can be deleted as they contain symbols in their lhs which do not occur in D:

+(x, 0) -> x
+(0, x) -> x

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 6`
`             ↳MRR`
`             ...`
`               →DP Problem 9`
`                 ↳Modular Removal of Rules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

We have the following set of usable rules:

+(x, +(y, z)) -> +(+(x, y), z)
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  1 + x1 + x2

We have the following set D of usable symbols: {+, +'}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 6`
`             ↳MRR`
`             ...`
`               →DP Problem 10`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳Usable Rules (Innermost)`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

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

As we are in the innermost case, we can delete all 36 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`           →DP Problem 11`
`             ↳Modular Removal of Rules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

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:

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

Strategy:

innermost

We have the following set of usable rules:

-(I(x), O(y)) -> I(-(x, y))
-(x, 0) -> x
-(I(x), I(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1)))
-(0, x) -> 0
-(O(x), O(y)) -> O(-(x, y))
O(0) -> 0
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(I(x1)) =  x1 POL(-'(x1, x2)) =  1 + x1 + x2 POL(0) =  1 POL(1) =  0 POL(O(x1)) =  x1 POL(-(x1, x2)) =  x1 + x2

We have the following set D of usable symbols: {I, -', 0, 1, O, -}
No Dependency Pairs can be deleted.
The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

-(x, 0) -> x

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`           →DP Problem 11`
`             ↳MRR`
`             ...`
`               →DP Problem 12`
`                 ↳Modular Removal of Rules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

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:

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

Strategy:

innermost

We have the following set of usable rules:

-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1)))
O(0) -> 0
-(0, x) -> 0
-(O(x), O(y)) -> O(-(x, y))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(I(x1)) =  1 + x1 POL(-'(x1, x2)) =  1 + x1 + x2 POL(0) =  0 POL(1) =  0 POL(O(x1)) =  1 + x1 POL(-(x1, x2)) =  x1 + x2

We have the following set D of usable symbols: {I, -', 0, 1, O, -}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

-'(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)

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳Usable Rules (Innermost)`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

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

As we are in the innermost case, we can delete all 43 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`           →DP Problem 13`
`             ↳Size-Change Principle`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. GE(0, O(x)) -> GE(0, x)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
O(x1) -> O(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳Usable Rules (Innermost)`
`       →DP Problem 5`
`         ↳UsableRules`

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

As we are in the innermost case, we can delete all 43 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`           →DP Problem 14`
`             ↳Size-Change Principle`
`       →DP Problem 5`
`         ↳UsableRules`

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)

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. GE(I(x), I(y)) -> GE(x, y)
2. GE(I(x), O(y)) -> GE(x, y)
3. GE(O(x), I(y)) -> GE(y, x)
4. GE(O(x), O(y)) -> GE(x, y)
and get the following Size-Change Graph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2>2
{1, 2, 3, 4} , {1, 2, 3, 4}
1>2
2>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
I(x1) -> I(x1)
O(x1) -> O(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳Usable Rules (Innermost)`

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

As we are in the innermost case, we can delete all 43 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`       →DP Problem 2`
`         ↳UsableRules`
`       →DP Problem 3`
`         ↳UsableRules`
`       →DP Problem 4`
`         ↳UsableRules`
`       →DP Problem 5`
`         ↳UsableRules`
`           →DP Problem 15`
`             ↳Size-Change Principle`

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. LOG'(O(x)) -> LOG'(x)
2. LOG'(I(x)) -> LOG'(x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
I(x1) -> I(x1)
O(x1) -> O(x1)

We obtain no new DP problems.

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