Term Rewriting System R:
[Y, X, X1, X2, X3]
aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

AMINUS(s(X), s(Y)) -> AMINUS(X, Y)
AGEQ(s(X), s(Y)) -> AGEQ(X, Y)
ADIV(s(X), s(Y)) -> AIF(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
AIF(true, X, Y) -> MARK(X)
AIF(false, X, Y) -> MARK(Y)
MARK(minus(X1, X2)) -> AMINUS(X1, X2)
MARK(geq(X1, X2)) -> AGEQ(X1, X2)
MARK(div(X1, X2)) -> MARK(X1)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains three SCCs.

R
DPs
→DP Problem 1
Polynomial Ordering
→DP Problem 2
Polo
→DP Problem 3
Polo

Dependency Pair:

AMINUS(s(X), s(Y)) -> AMINUS(X, Y)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The following dependency pair can be strictly oriented:

AMINUS(s(X), s(Y)) -> AMINUS(X, Y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(A__MINUS(x1, x2)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 4
Dependency Graph
→DP Problem 2
Polo
→DP Problem 3
Polo

Dependency Pair:

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polynomial Ordering
→DP Problem 3
Polo

Dependency Pair:

AGEQ(s(X), s(Y)) -> AGEQ(X, Y)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The following dependency pair can be strictly oriented:

AGEQ(s(X), s(Y)) -> AGEQ(X, Y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(A__GEQ(x1, x2)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 5
Dependency Graph
→DP Problem 3
Polo

Dependency Pair:

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polynomial Ordering

Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
AIF(false, X, Y) -> MARK(Y)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), X2, X3)
MARK(div(X1, X2)) -> MARK(X1)
AIF(true, X, Y) -> MARK(X)
ADIV(s(X), s(Y)) -> AIF(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(if(X1, X2, X3)) -> AIF(mark(X1), X2, X3)

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(geq(x1, x2)) =  0 POL(false) =  0 POL(minus(x1, x2)) =  0 POL(true) =  0 POL(mark(x1)) =  0 POL(a__div(x1, x2)) =  0 POL(if(x1, x2, x3)) =  1 + x1 + x2 + x3 POL(0) =  0 POL(a__if(x1, x2, x3)) =  0 POL(a__minus(x1, x2)) =  0 POL(s(x1)) =  x1 POL(a__geq(x1, x2)) =  0 POL(div(x1, x2)) =  x1 POL(A__DIV(x1, x2)) =  0 POL(A__IF(x1, x2, x3)) =  x2 + x3

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 6
Polynomial Ordering

Dependency Pairs:

MARK(s(X)) -> MARK(X)
AIF(false, X, Y) -> MARK(Y)
MARK(div(X1, X2)) -> MARK(X1)
AIF(true, X, Y) -> MARK(X)
ADIV(s(X), s(Y)) -> AIF(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(div(X1, X2)) -> MARK(X1)

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(geq(x1, x2)) =  0 POL(false) =  0 POL(minus(x1, x2)) =  0 POL(true) =  0 POL(mark(x1)) =  0 POL(a__div(x1, x2)) =  0 POL(if(x1, x2, x3)) =  0 POL(0) =  0 POL(a__if(x1, x2, x3)) =  0 POL(a__minus(x1, x2)) =  0 POL(s(x1)) =  x1 POL(a__geq(x1, x2)) =  0 POL(div(x1, x2)) =  1 + x1 POL(A__DIV(x1, x2)) =  1 POL(A__IF(x1, x2, x3)) =  x2 + x3

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 6
Polo
...
→DP Problem 7
Polynomial Ordering

Dependency Pairs:

MARK(s(X)) -> MARK(X)
AIF(false, X, Y) -> MARK(Y)
AIF(true, X, Y) -> MARK(X)
ADIV(s(X), s(Y)) -> AIF(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(s(X)) -> MARK(X)

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

aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(geq(x1, x2)) =  0 POL(false) =  0 POL(minus(x1, x2)) =  0 POL(true) =  0 POL(mark(x1)) =  x1 POL(a__div(x1, x2)) =  x1 POL(if(x1, x2, x3)) =  x2 + x3 POL(0) =  0 POL(a__if(x1, x2, x3)) =  x2 + x3 POL(a__minus(x1, x2)) =  0 POL(s(x1)) =  1 + x1 POL(a__geq(x1, x2)) =  0 POL(div(x1, x2)) =  x1 POL(A__DIV(x1, x2)) =  x1 POL(A__IF(x1, x2, x3)) =  x2 + x3

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 6
Polo
...
→DP Problem 8
Polynomial Ordering

Dependency Pairs:

AIF(false, X, Y) -> MARK(Y)
AIF(true, X, Y) -> MARK(X)
ADIV(s(X), s(Y)) -> AIF(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(geq(x1, x2)) =  0 POL(false) =  0 POL(minus(x1, x2)) =  0 POL(true) =  0 POL(mark(x1)) =  0 POL(a__div(x1, x2)) =  0 POL(if(x1, x2, x3)) =  0 POL(0) =  0 POL(a__if(x1, x2, x3)) =  0 POL(a__minus(x1, x2)) =  0 POL(s(x1)) =  0 POL(a__geq(x1, x2)) =  0 POL(div(x1, x2)) =  1 POL(A__DIV(x1, x2)) =  0 POL(A__IF(x1, x2, x3)) =  x2 + x3

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 6
Polo
...
→DP Problem 9
Dependency Graph

Dependency Pairs:

AIF(false, X, Y) -> MARK(Y)
AIF(true, X, Y) -> MARK(X)
ADIV(s(X), s(Y)) -> AIF(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)

Rules:

aminus(0, Y) -> 0
aminus(s(X), s(Y)) -> aminus(X, Y)
aminus(X1, X2) -> minus(X1, X2)
ageq(X, 0) -> true
ageq(0, s(Y)) -> false
ageq(s(X), s(Y)) -> ageq(X, Y)
ageq(X1, X2) -> geq(X1, X2)
adiv(s(X), s(Y)) -> aif(ageq(X, Y), s(div(minus(X, Y), s(Y))), 0)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> aminus(X1, X2)
mark(geq(X1, X2)) -> ageq(X1, X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes