Term Rewriting System R:
[x, y]
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

MIN(s(x), s(y)) -> MIN(x, y)
MAX(s(x), s(y)) -> MAX(x, y)
-'(s(x), s(y)) -> -'(x, y)
GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
GCD(s(x), s(y)) -> -'(s(max(x, y)), s(min(x, y)))
GCD(s(x), s(y)) -> MAX(x, y)
GCD(s(x), s(y)) -> MIN(x, y)

Furthermore, R contains four SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Remaining


Dependency Pair:

MIN(s(x), s(y)) -> MIN(x, y)


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MIN(s(x), s(y)) -> MIN(x, y)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
MIN(x1, x2) -> MIN(x1, x2)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 5
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Remaining


Dependency Pair:


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Argument Filtering and Ordering
       →DP Problem 3
AFS
       →DP Problem 4
Remaining


Dependency Pair:

MAX(s(x), s(y)) -> MAX(x, y)


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MAX(s(x), s(y)) -> MAX(x, y)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
MAX(x1, x2) -> MAX(x1, x2)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 6
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
Remaining


Dependency Pair:


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Argument Filtering and Ordering
       →DP Problem 4
Remaining


Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

-'(s(x), s(y)) -> -'(x, y)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
-'(x1, x2) -> -'(x1, x2)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 7
Dependency Graph
       →DP Problem 4
Remaining


Dependency Pair:


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Rules:


min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:02 minutes