Term Rewriting System R:
[x, y]
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(x, s(y)) -> +'(x, y)
*'(x, s(y)) -> +'(*(x, y), x)
*'(x, s(y)) -> *'(x, y)
GE(s(x), s(y)) -> GE(x, y)
-'(s(x), s(y)) -> -'(x, y)
FACT(x) -> IFFACT(x, ge(x, s(s(0))))
FACT(x) -> GE(x, s(s(0)))
IFFACT(x, true) -> *'(x, fact(-(x, s(0))))
IFFACT(x, true) -> FACT(-(x, s(0)))
IFFACT(x, true) -> -'(x, s(0))

Furthermore, R contains five SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 6
Dependency Graph
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


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
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(GE(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 7
Dependency Graph
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


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
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(-'(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 3
Polo
           →DP Problem 8
Dependency Graph
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


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
Polo
       →DP Problem 4
Polynomial Ordering
       →DP Problem 5
Nar


Dependency Pair:

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


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
           →DP Problem 9
Dependency Graph
       →DP Problem 5
Nar


Dependency Pair:


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


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
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Narrowing Transformation


Dependency Pairs:

IFFACT(x, true) -> FACT(-(x, s(0)))
FACT(x) -> IFFACT(x, ge(x, s(s(0))))


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




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

FACT(x) -> IFFACT(x, ge(x, s(s(0))))
two new Dependency Pairs are created:

FACT(0) -> IFFACT(0, false)
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar
           →DP Problem 10
Narrowing Transformation


Dependency Pairs:

FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))
IFFACT(x, true) -> FACT(-(x, s(0)))


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




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

IFFACT(x, true) -> FACT(-(x, s(0)))
one new Dependency Pair is created:

IFFACT(s(x''), true) -> FACT(-(x'', 0))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar
           →DP Problem 10
Nar
             ...
               →DP Problem 11
Rewriting Transformation


Dependency Pairs:

IFFACT(s(x''), true) -> FACT(-(x'', 0))
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




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

IFFACT(s(x''), true) -> FACT(-(x'', 0))
one new Dependency Pair is created:

IFFACT(s(x''), true) -> FACT(x'')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar
           →DP Problem 10
Nar
             ...
               →DP Problem 12
Polynomial Ordering


Dependency Pairs:

IFFACT(s(x''), true) -> FACT(x'')
FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

IFFACT(s(x''), true) -> FACT(x'')


Additionally, the following usable rules for innermost can be oriented:

ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(FACT(x1))=  x1  
  POL(false)=  0  
  POL(IFFACT(x1, x2))=  x1  
  POL(true)=  0  
  POL(s(x1))=  1 + x1  
  POL(ge(x1, x2))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar
           →DP Problem 10
Nar
             ...
               →DP Problem 13
Dependency Graph


Dependency Pair:

FACT(s(x'')) -> IFFACT(s(x''), ge(x'', s(0)))


Rules:


+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
ge(x, 0) -> true
ge(0, s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0))))
iffact(x, true) -> *(x, fact(-(x, s(0))))
iffact(x, false) -> s(0)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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