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)))

Innermost 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

         ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

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)))

Strategy:

innermost

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

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

seven new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 10

             ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

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

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

seven new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

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

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost
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)))

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:10 minutes