Term Rewriting System R:
[x, y, z]
O(0) -> 0
+(0, x) -> x
+(x, 0) -> x
+(O(x), O(y)) -> O(+(x, y))
+(O(x), I(y)) -> I(+(x, y))
+(I(x), O(y)) -> I(+(x, y))
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
+(x, +(y, z)) -> +(+(x, y), z)
-(x, 0) -> x
-(0, x) -> 0
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1)))
-(I(x), O(y)) -> I(-(x, y))
-(I(x), I(y)) -> O(-(x, y))
not(true) -> false
not(false) -> true
and(x, true) -> x
and(x, false) -> false
if(true, x, y) -> x
if(false, x, y) -> y
ge(O(x), O(y)) -> ge(x, y)
ge(O(x), I(y)) -> not(ge(y, x))
ge(I(x), O(y)) -> ge(x, y)
ge(I(x), I(y)) -> ge(x, y)
ge(x, 0) -> true
ge(0, O(x)) -> ge(0, x)
ge(0, I(x)) -> false
Log'(0) -> 0
Log'(I(x)) -> +(Log'(x), I(0))
Log'(O(x)) -> if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) -> -(Log'(x), I(0))
Val(L(x)) -> x
Val(N(x, l, r)) -> x
Min(L(x)) -> x
Min(N(x, l, r)) -> Min(l)
Max(L(x)) -> x
Max(N(x, l, r)) -> Max(r)
BS(L(x)) -> true
BS(N(x, l, r)) -> and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) -> I(0)
Size(N(x, l, r)) -> +(+(Size(l), Size(r)), I(1))
WB(L(x)) -> true
WB(N(x, l, r)) -> and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains five SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

+'(I(x), I(y)) -> +'(+(x, y), I(0))
seven new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 6
             ↳Narrowing Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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:

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

The transformation is resulting in one new DP problem: 



   R
     ↳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)




The following remains to be proven:


   R
     ↳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)




The following remains to be proven:


   R
     ↳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)




The following remains to be proven:


   R
     ↳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)




The following remains to be proven:


   R
     ↳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)




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:07 minutes