YES

The TRS could be proven terminating. The proof took 2350 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (878ms).
 | – Problem 2 was processed with processor SubtermCriterion (9ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4iUR (100ms).
 |    | – Problem 7 was processed with processor PolynomialLinearRange4iUR (16ms).
 | – Problem 4 was processed with processor PolynomialLinearRange4iUR (398ms).
 |    | – Problem 8 was processed with processor PolynomialLinearRange4iUR (187ms).
 |    |    | – Problem 10 was processed with processor PolynomialLinearRange4iUR (196ms).
 |    |    |    | – Problem 12 was processed with processor PolynomialLinearRange4iUR (83ms).
 | – Problem 5 was processed with processor SubtermCriterion (0ms).
 | – Problem 6 was processed with processor PolynomialLinearRange4iUR (153ms).
 |    | – Problem 9 was processed with processor PolynomialLinearRange4iUR (102ms).
 |    |    | – Problem 11 was processed with processor PolynomialLinearRange4iUR (88ms).

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

+#(O(x), O(y)) → O#(+(x, y))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)) → ge#(x, Max(l))ge#(I(x), O(y)) → ge#(x, y)
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)))Log'#(I(x)) → +#(Log'(x), I(0))
-#(O(x), O(y)) → -#(x, y)-#(I(x), I(y)) → -#(x, y)
BS#(N(x, l, r)) → BS#(l)Log'#(O(x)) → ge#(x, I(0))
WB#(N(x, l, r)) → Size#(r)ge#(O(x), I(y)) → not#(ge(y, x))
WB#(N(x, l, r)) → WB#(r)+#(O(x), O(y)) → +#(x, y)
+#(I(x), I(y)) → O#(+(+(x, y), I(0)))Log'#(I(x)) → Log'#(x)
BS#(N(x, l, r)) → and#(BS(l), BS(r))Log#(x) → Log'#(x)
WB#(N(x, l, r)) → ge#(I(0), -(Size(r), Size(l)))WB#(N(x, l, r)) → ge#(I(0), -(Size(l), Size(r)))
Size#(N(x, l, r)) → +#(Size(l), Size(r))Size#(N(x, l, r)) → +#(+(Size(l), Size(r)), I(1))
BS#(N(x, l, r)) → Min#(r)Log'#(O(x)) → Log'#(x)
Log'#(O(x)) → +#(Log'(x), I(0))+#(x, +(y, z)) → +#(x, y)
WB#(N(x, l, r)) → Size#(l)WB#(N(x, l, r)) → and#(WB(l), WB(r))
ge#(I(x), I(y)) → ge#(x, y)ge#(O(x), O(y)) → ge#(x, y)
-#(O(x), I(y)) → -#(x, y)WB#(N(x, l, r)) → -#(Size(r), Size(l))
Size#(N(x, l, r)) → Size#(l)Min#(N(x, l, r)) → Min#(l)
BS#(N(x, l, r)) → Max#(l)BS#(N(x, l, r)) → and#(ge(x, Max(l)), ge(Min(r), x))
+#(I(x), I(y)) → +#(x, y)+#(I(x), O(y)) → +#(x, y)
+#(O(x), I(y)) → +#(x, y)+#(I(x), I(y)) → +#(+(x, y), I(0))
WB#(N(x, l, r)) → ge#(Size(l), Size(r))ge#(0, O(x)) → ge#(0, x)
-#(I(x), I(y)) → O#(-(x, y))Max#(N(x, l, r)) → Max#(r)
-#(I(x), O(y)) → -#(x, y)WB#(N(x, l, r)) → WB#(l)
Log#(x) → -#(Log'(x), I(0))+#(x, +(y, z)) → +#(+(x, y), z)
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))))BS#(N(x, l, r)) → BS#(r)
WB#(N(x, l, r)) → -#(Size(l), Size(r))-#(O(x), O(y)) → O#(-(x, y))
ge#(O(x), I(y)) → ge#(y, x)BS#(N(x, l, r)) → ge#(Min(r), x)
-#(O(x), I(y)) → -#(-(x, y), I(1))Size#(N(x, l, r)) → Size#(r)
Log'#(O(x)) → if#(ge(x, I(0)), +(Log'(x), I(0)), 0)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

The following SCCs where found

+#(I(x), I(y)) → +#(x, y)+#(I(x), O(y)) → +#(x, y)
+#(O(x), I(y)) → +#(x, y)+#(x, +(y, z)) → +#(+(x, y), z)
+#(I(x), I(y)) → +#(+(x, y), I(0))+#(O(x), O(y)) → +#(x, y)
+#(x, +(y, z)) → +#(x, y)
Log'#(O(x)) → Log'#(x)Log'#(I(x)) → Log'#(x)
-#(I(x), O(y)) → -#(x, y)-#(O(x), I(y)) → -#(-(x, y), I(1))
-#(O(x), O(y)) → -#(x, y)-#(I(x), I(y)) → -#(x, y)
-#(O(x), I(y)) → -#(x, y)
ge#(0, O(x)) → ge#(0, 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#(O(x), O(y)) → ge#(x, y)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

ge#(0, O(x)) → ge#(0, x)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

ge#(0, O(x)) → ge#(0, x)

Problem 3: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

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#(O(x), O(y)) → ge#(x, y)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

ge#(O(x), I(y)) → ge#(y, x)ge#(O(x), O(y)) → ge#(x, y)
ge#(I(x), O(y)) → ge#(x, y)

Problem 7: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

ge#(I(x), I(y)) → ge#(x, y)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, WB, 1, Log, 0, r, if, Max, false, Val, Min, Size

Strategy

Polynomial Interpretation

There are no usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

ge#(I(x), I(y)) → ge#(x, y)

Problem 4: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

+#(I(x), I(y)) → +#(x, y)+#(I(x), O(y)) → +#(x, y)
+#(O(x), I(y)) → +#(x, y)+#(x, +(y, z)) → +#(+(x, y), z)
+#(I(x), I(y)) → +#(+(x, y), I(0))+#(O(x), O(y)) → +#(x, y)
+#(x, +(y, z)) → +#(x, y)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

+#(I(x), O(y)) → +#(x, y)+#(O(x), O(y)) → +#(x, y)

Problem 8: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

+#(I(x), I(y)) → +#(x, y)+#(O(x), I(y)) → +#(x, y)
+#(I(x), I(y)) → +#(+(x, y), I(0))+#(x, +(y, z)) → +#(+(x, y), z)
+#(x, +(y, z)) → +#(x, y)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, WB, 1, Log, 0, r, if, Max, false, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

+#(x, +(y, z)) → +#(+(x, y), z)+#(x, +(y, z)) → +#(x, y)

Problem 10: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

+#(I(x), I(y)) → +#(x, y)+#(O(x), I(y)) → +#(x, y)
+#(I(x), I(y)) → +#(+(x, y), I(0))

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

+#(I(x), I(y)) → +#(x, y)+#(O(x), I(y)) → +#(x, y)

Problem 12: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

+#(I(x), I(y)) → +#(+(x, y), I(0))

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, WB, 1, Log, 0, r, if, Max, false, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

+(x, +(y, z)) → +(+(x, y), z)O(0) → 0
+(O(x), O(y)) → O(+(x, y))+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(0, x) → x+(I(x), O(y)) → I(+(x, y))
+(O(x), I(y)) → I(+(x, y))+(x, 0) → x

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

+#(I(x), I(y)) → +#(+(x, y), I(0))

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

Log'#(O(x)) → Log'#(x)Log'#(I(x)) → Log'#(x)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

Log'#(O(x)) → Log'#(x)Log'#(I(x)) → Log'#(x)

Problem 6: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

-#(I(x), O(y)) → -#(x, y)-#(O(x), I(y)) → -#(-(x, y), I(1))
-#(O(x), O(y)) → -#(x, y)-#(I(x), I(y)) → -#(x, y)
-#(O(x), I(y)) → -#(x, y)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

-#(I(x), O(y)) → -#(x, y)-#(O(x), O(y)) → -#(x, y)

Problem 9: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

-#(O(x), I(y)) → -#(-(x, y), I(1))-#(I(x), I(y)) → -#(x, y)
-#(O(x), I(y)) → -#(x, y)

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, WB, 1, Log, 0, r, if, Max, false, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

-#(I(x), I(y)) → -#(x, y)-#(O(x), I(y)) → -#(x, y)

Problem 11: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

-#(O(x), I(y)) → -#(-(x, y), I(1))

Rewrite 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) → falsenot(false) → true
and(x, true) → xand(x, false) → false
if(true, x, y) → xif(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) → truege(0, O(x)) → ge(0, x)
ge(0, I(x)) → falseLog'(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)) → xMin(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)))

Original Signature

Termination of terms over the following signature is verified: BS, L, +, true, Log', N, l, ge, O, I, -, and, not, 1, WB, 0, Log, r, if, false, Max, Val, Min, Size

Strategy

Polynomial Interpretation

Improved Usable rules

-(I(x), I(y)) → O(-(x, y))-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(x, 0) → xO(0) → 0
-(I(x), O(y)) → I(-(x, y))-(O(x), O(y)) → O(-(x, y))
-(0, x) → 0

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

-#(O(x), I(y)) → -#(-(x, y), I(1))