Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

+'(0(x), 0(y)) -> 0'(+(x, y))
+'(0(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(1(x), 1(y)) -> 0'(+(+(x, y), 1(#)))
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 1(y)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)
-'(0(x), 0(y)) -> 0'(-(x, y))
-'(0(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
-'(0(x), 1(y)) -> -'(x, y)
-'(1(x), 0(y)) -> -'(x, y)
-'(1(x), 1(y)) -> 0'(-(x, y))
-'(1(x), 1(y)) -> -'(x, y)
GE(0(x), 0(y)) -> GE(x, y)
GE(0(x), 1(y)) -> NOT(ge(y, x))
GE(0(x), 1(y)) -> GE(y, x)
GE(1(x), 0(y)) -> GE(x, y)
GE(1(x), 1(y)) -> GE(x, y)
GE(#, 0(x)) -> GE(#, x)
MIN(n(x, y, z)) -> MIN(y)
MAX(n(x, y, z)) -> MAX(z)
BS(n(x, y, z)) -> AND(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
BS(n(x, y, z)) -> AND(ge(x, max(y)), ge(min(z), x))
BS(n(x, y, z)) -> GE(x, max(y))
BS(n(x, y, z)) -> MAX(y)
BS(n(x, y, z)) -> GE(min(z), x)
BS(n(x, y, z)) -> MIN(z)
BS(n(x, y, z)) -> AND(bs(y), bs(z))
BS(n(x, y, z)) -> BS(y)
BS(n(x, y, z)) -> BS(z)
SIZE(n(x, y, z)) -> +'(+(size(x), size(y)), 1(#))
SIZE(n(x, y, z)) -> +'(size(x), size(y))
SIZE(n(x, y, z)) -> SIZE(x)
SIZE(n(x, y, z)) -> SIZE(y)
WB(n(x, y, z)) -> AND(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))
WB(n(x, y, z)) -> IF(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y))))
WB(n(x, y, z)) -> GE(size(y), size(z))
WB(n(x, y, z)) -> SIZE(y)
WB(n(x, y, z)) -> SIZE(z)
WB(n(x, y, z)) -> GE(1(#), -(size(y), size(z)))
WB(n(x, y, z)) -> -'(size(y), size(z))
WB(n(x, y, z)) -> GE(1(#), -(size(z), size(y)))
WB(n(x, y, z)) -> -'(size(z), size(y))
WB(n(x, y, z)) -> AND(wb(y), wb(z))
WB(n(x, y, z)) -> WB(y)
WB(n(x, y, z)) -> WB(z)

Furthermore, R contains nine SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(0(x), 0(y)) -> +'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(#) = 0

POL(0(x₁)) = x₁

POL(1(x₁)) = x₁

POL(+(x₁, x₂)) = 1 + x₁ + x₂

POL(+'(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 10

             ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(0(x), 0(y)) -> +'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

+'(1(x), 1(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(#) = 0

POL(0(x₁)) = x₁

POL(1(x₁)) = 1 + x₁

POL(+(x₁, x₂)) = 0

POL(+'(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 10

             ↳Polo

...

               →DP Problem 11

                 ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 0(y)) -> +'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph the DP problem was split into 2 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 10

             ↳Polo

...

               →DP Problem 12

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

+'(1(x), 1(y)) -> +'(+(x, y), 1(#))

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pair can be strictly oriented:

+'(1(x), 1(y)) -> +'(+(x, y), 1(#))

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

+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
0(#) -> #

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(#) = 0

POL(0(x₁)) = x₁

POL(1(x₁)) = 1 + x₁

POL(+(x₁, x₂)) = x₁ + x₂

POL(+'(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 10

             ↳Polo

...

               →DP Problem 14

                 ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 10

             ↳Polo

...

               →DP Problem 13

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

+'(0(x), 0(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

+'(0(x), 0(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(0(x₁)) = 1 + x₁

POL(1(x₁)) = 0

POL(+'(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

-'(1(x), 1(y)) -> -'(x, y)
-'(1(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
-'(0(x), 0(y)) -> -'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(#) = 0

POL(0(x₁)) = x₁

POL(-'(x₁, x₂)) = x₂

POL(1(x₁)) = 1 + x₁

POL(-(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 16

             ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

-'(1(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
-'(0(x), 0(y)) -> -'(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph the DP problem was split into 2 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 16

             ↳DGraph

...

               →DP Problem 17

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

-'(0(x), 1(y)) -> -'(-(x, y), 1(#))

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pair can be strictly oriented:

-'(0(x), 1(y)) -> -'(-(x, y), 1(#))

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

-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
0(#) -> #

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(#) = 0

POL(0(x₁)) = 1 + x₁

POL(-'(x₁, x₂)) = 1 + x₁

POL(1(x₁)) = 1 + x₁

POL(-(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 16

             ↳DGraph

...

               →DP Problem 19

                 ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 16

             ↳DGraph

...

               →DP Problem 18

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

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

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(-'(x₁, x₂)) = x₂

POL(0(x₁)) = 1 + x₁

POL(1(x₁)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

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

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(#) = 0

POL(0(x₁)) = 1 + x₁

POL(GE(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 21

             ↳Dependency Graph

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

MIN(n(x, y, z)) -> MIN(y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pair can be strictly oriented:

MIN(n(x, y, z)) -> MIN(y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MIN(x₁)) = x₁

POL(n(x₁, x₂, x₃)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 22

             ↳Dependency Graph

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

MAX(n(x, y, z)) -> MAX(z)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pair can be strictly oriented:

MAX(n(x, y, z)) -> MAX(z)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MAX(x₁)) = x₁

POL(n(x₁, x₂, x₃)) = 1 + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 23

             ↳Dependency Graph

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polynomial Ordering

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

SIZE(n(x, y, z)) -> SIZE(y)
SIZE(n(x, y, z)) -> SIZE(x)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

SIZE(n(x, y, z)) -> SIZE(y)
SIZE(n(x, y, z)) -> SIZE(x)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n(x₁, x₂, x₃)) = 1 + x₁ + x₂

POL(SIZE(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

           →DP Problem 24

             ↳Dependency Graph

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polynomial Ordering

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pairs:

GE(1(x), 1(y)) -> GE(x, y)
GE(1(x), 0(y)) -> GE(x, y)
GE(0(x), 1(y)) -> GE(y, x)
GE(0(x), 0(y)) -> GE(x, y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

GE(1(x), 1(y)) -> GE(x, y)
GE(1(x), 0(y)) -> GE(x, y)
GE(0(x), 1(y)) -> GE(y, x)
GE(0(x), 0(y)) -> GE(x, y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(0(x₁)) = 1 + x₁

POL(GE(x₁, x₂)) = 1 + x₁ + x₂

POL(1(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

           →DP Problem 25

             ↳Dependency Graph

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polynomial Ordering

       →DP Problem 9

         ↳Polo

Dependency Pairs:

WB(n(x, y, z)) -> WB(z)
WB(n(x, y, z)) -> WB(y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

WB(n(x, y, z)) -> WB(z)
WB(n(x, y, z)) -> WB(y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n(x₁, x₂, x₃)) = 1 + x₂ + x₃

POL(WB(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

           →DP Problem 26

             ↳Dependency Graph

       →DP Problem 9

         ↳Polo

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polynomial Ordering

Dependency Pairs:

BS(n(x, y, z)) -> BS(z)
BS(n(x, y, z)) -> BS(y)

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

The following dependency pairs can be strictly oriented:

BS(n(x, y, z)) -> BS(z)
BS(n(x, y, z)) -> BS(y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(BS(x₁)) = x₁

POL(n(x₁, x₂, x₃)) = 1 + x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

           →DP Problem 27

             ↳Dependency Graph

Dependency Pair:

Rules:

0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, #) -> x
-(#, x) -> #
-(0(x), 0(y)) -> 0(-(x, y))
-(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) -> 1(-(x, y))
-(1(x), 1(y)) -> 0(-(x, y))
not(false) -> true
not(true) -> false
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(0(x), 0(y)) -> ge(x, y)
ge(0(x), 1(y)) -> not(ge(y, x))
ge(1(x), 0(y)) -> ge(x, y)
ge(1(x), 1(y)) -> ge(x, y)
ge(x, #) -> true
ge(#, 1(x)) -> false
ge(#, 0(x)) -> ge(#, x)
val(l(x)) -> x
val(n(x, y, z)) -> x
min(l(x)) -> x
min(n(x, y, z)) -> min(y)
max(l(x)) -> x
max(n(x, y, z)) -> max(z)
bs(l(x)) -> true
bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
size(l(x)) -> 1(#)
size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
wb(l(x)) -> true
wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes

POL(#)	= 0
POL(0(x₁))	= x₁
POL(1(x₁))	= x₁
POL(+(x₁, x₂))	= 1 + x₁ + x₂
POL(+'(x₁, x₂))	= x₂

POL(0(x₁))	= 1 + x₁
POL(1(x₁))	= 0
POL(+'(x₁, x₂))	= x₂

POL(-'(x₁, x₂))	= x₂
POL(0(x₁))	= 1 + x₁
POL(1(x₁))	= 0

POL(MIN(x₁))	= x₁
POL(n(x₁, x₂, x₃))	= 1 + x₂

POL(MAX(x₁))	= x₁
POL(n(x₁, x₂, x₃))	= 1 + x₃

POL(n(x₁, x₂, x₃))	= 1 + x₁ + x₂
POL(SIZE(x₁))	= x₁

POL(0(x₁))	= 1 + x₁
POL(GE(x₁, x₂))	= 1 + x₁ + x₂
POL(1(x₁))	= 1 + x₁

POL(n(x₁, x₂, x₃))	= 1 + x₂ + x₃
POL(WB(x₁))	= x₁

POL(BS(x₁))	= x₁
POL(n(x₁, x₂, x₃))	= 1 + x₂ + x₃