Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
LT(s(x), s(y)) -> LT(x, y)
DIV(s(x), s(y)) -> IF(lt(x, y), 0, s(div(-(x, y), s(y))))
DIV(s(x), s(y)) -> LT(x, y)
DIV(s(x), s(y)) -> DIV(-(x, y), s(y))
DIV(s(x), s(y)) -> -'(x, y)

Furthermore, R contains three SCCs.


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


Dependency Pair:

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


Rules:


-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement 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 4
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS


Dependency Pair:


Rules:


-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))





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


Dependency Pair:

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


Rules:


-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement 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:
LT(x1, x2) -> LT(x1, x2)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))





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


Dependency Pair:

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


Rules:


-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))





The following dependency pair can be strictly oriented:

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


The following usable rules using the Ce-refinement can be oriented:

-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


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


Dependency Pair:


Rules:


-(x, 0) -> x
-(0, s(y)) -> 0
-(s(x), s(y)) -> -(x, y)
lt(x, 0) -> false
lt(0, s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0) -> 0
div(0, y) -> 0
div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes