Term Rewriting System R:
[X, Y]
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

MINUS(X, s(Y)) -> PRED(minus(X, Y))
MINUS(X, s(Y)) -> MINUS(X, Y)
LE(s(X), s(Y)) -> LE(X, Y)
GCD(s(X), s(Y)) -> IF(le(Y, X), s(X), s(Y))
GCD(s(X), s(Y)) -> LE(Y, X)
IF(true, s(X), s(Y)) -> GCD(minus(X, Y), s(Y))
IF(true, s(X), s(Y)) -> MINUS(X, Y)
IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))
IF(false, s(X), s(Y)) -> MINUS(Y, X)

Furthermore, R contains three SCCs.


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


Dependency Pair:

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


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
MINUS(x1, x2) -> MINUS(x1, x2)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


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
Nar


Dependency Pair:

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


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
LE(x1, x2) -> LE(x1, x2)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


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
Narrowing Transformation


Dependency Pairs:

IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))
IF(true, s(X), s(Y)) -> GCD(minus(X, Y), s(Y))
GCD(s(X), s(Y)) -> IF(le(Y, X), s(X), s(Y))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




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

GCD(s(X), s(Y)) -> IF(le(Y, X), s(X), s(Y))
three new Dependency Pairs are created:

GCD(s(s(Y'')), s(s(X''))) -> IF(le(X'', Y''), s(s(Y'')), s(s(X'')))
GCD(s(0), s(s(X''))) -> IF(false, s(0), s(s(X'')))
GCD(s(X'), s(0)) -> IF(true, s(X'), s(0))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Nar
           →DP Problem 6
Narrowing Transformation


Dependency Pairs:

GCD(s(X'), s(0)) -> IF(true, s(X'), s(0))
GCD(s(0), s(s(X''))) -> IF(false, s(0), s(s(X'')))
IF(true, s(X), s(Y)) -> GCD(minus(X, Y), s(Y))
GCD(s(s(Y'')), s(s(X''))) -> IF(le(X'', Y''), s(s(Y'')), s(s(X'')))
IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




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

IF(true, s(X), s(Y)) -> GCD(minus(X, Y), s(Y))
two new Dependency Pairs are created:

IF(true, s(X''), s(s(Y''))) -> GCD(pred(minus(X'', Y'')), s(s(Y'')))
IF(true, s(X''), s(0)) -> GCD(X'', s(0))

The transformation is resulting in two new DP problems:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 7
Argument Filtering and Ordering


Dependency Pairs:

IF(true, s(X''), s(0)) -> GCD(X'', s(0))
GCD(s(X'), s(0)) -> IF(true, s(X'), s(0))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

IF(true, s(X''), s(0)) -> GCD(X'', s(0))


There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
IF(x1, x2, x3) -> x2
s(x1) -> s(x1)
GCD(x1, x2) -> x1


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 12
Dependency Graph


Dependency Pair:

GCD(s(X'), s(0)) -> IF(true, s(X'), s(0))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


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
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 8
Narrowing Transformation


Dependency Pairs:

IF(true, s(X''), s(s(Y''))) -> GCD(pred(minus(X'', Y'')), s(s(Y'')))
GCD(s(s(Y'')), s(s(X''))) -> IF(le(X'', Y''), s(s(Y'')), s(s(X'')))
IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))
GCD(s(0), s(s(X''))) -> IF(false, s(0), s(s(X'')))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




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

IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))
two new Dependency Pairs are created:

IF(false, s(s(Y'')), s(Y0)) -> GCD(pred(minus(Y0, Y'')), s(s(Y'')))
IF(false, s(0), s(Y')) -> GCD(Y', s(0))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 9
Instantiation Transformation


Dependency Pairs:

IF(false, s(s(Y'')), s(Y0)) -> GCD(pred(minus(Y0, Y'')), s(s(Y'')))
GCD(s(s(Y'')), s(s(X''))) -> IF(le(X'', Y''), s(s(Y'')), s(s(X'')))
IF(true, s(X''), s(s(Y''))) -> GCD(pred(minus(X'', Y'')), s(s(Y'')))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




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

IF(true, s(X''), s(s(Y''))) -> GCD(pred(minus(X'', Y'')), s(s(Y'')))
one new Dependency Pair is created:

IF(true, s(s(Y''''')), s(s(Y''''))) -> GCD(pred(minus(s(Y'''''), Y'''')), s(s(Y'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 10
Instantiation Transformation


Dependency Pairs:

IF(true, s(s(Y''''')), s(s(Y''''))) -> GCD(pred(minus(s(Y'''''), Y'''')), s(s(Y'''')))
GCD(s(s(Y'')), s(s(X''))) -> IF(le(X'', Y''), s(s(Y'')), s(s(X'')))
IF(false, s(s(Y'')), s(Y0)) -> GCD(pred(minus(Y0, Y'')), s(s(Y'')))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost




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

IF(false, s(s(Y'')), s(Y0)) -> GCD(pred(minus(Y0, Y'')), s(s(Y'')))
one new Dependency Pair is created:

IF(false, s(s(Y'''')), s(s(X''''))) -> GCD(pred(minus(s(X''''), Y'''')), s(s(Y'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 11
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

IF(false, s(s(Y'''')), s(s(X''''))) -> GCD(pred(minus(s(X''''), Y'''')), s(s(Y'''')))
GCD(s(s(Y'')), s(s(X''))) -> IF(le(X'', Y''), s(s(Y'')), s(s(X'')))
IF(true, s(s(Y''''')), s(s(Y''''))) -> GCD(pred(minus(s(Y'''''), Y'''')), s(s(Y'''')))


Rules:


minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0) -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0) -> false
le(0, Y) -> true
gcd(0, Y) -> 0
gcd(s(X), 0) -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))


Strategy:

innermost



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