Term Rewriting System R:
[x, y]
p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

P(p(s(x))) -> P(x)
LE(p(s(x)), x) -> LE(x, x)
LE(s(x), s(y)) -> LE(x, y)
MINUS(x, y) -> IF(le(x, y), x, y)
MINUS(x, y) -> LE(x, y)
IF(false, x, y) -> MINUS(p(x), y)
IF(false, x, y) -> P(x)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


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 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:
LE(x1, x2) -> LE(x1, x2)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Narrowing Transformation


Dependency Pairs:

IF(false, x, y) -> MINUS(p(x), y)
MINUS(x, y) -> IF(le(x, y), x, y)


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

MINUS(x, y) -> IF(le(x, y), x, y)
three new Dependency Pairs are created:

MINUS(0, y'') -> IF(true, 0, y'')
MINUS(s(x''), 0) -> IF(false, s(x''), 0)
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Narrowing Transformation


Dependency Pairs:

MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))
MINUS(s(x''), 0) -> IF(false, s(x''), 0)
IF(false, x, y) -> MINUS(p(x), y)


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

IF(false, x, y) -> MINUS(p(x), y)
two new Dependency Pairs are created:

IF(false, 0, y) -> MINUS(s(s(0)), y)
IF(false, s(x''), y) -> MINUS(x'', y)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 5
                 ↳Instantiation Transformation


Dependency Pairs:

MINUS(s(x''), 0) -> IF(false, s(x''), 0)
IF(false, s(x''), y) -> MINUS(x'', y)
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

IF(false, s(x''), y) -> MINUS(x'', y)
two new Dependency Pairs are created:

IF(false, s(x''''), 0) -> MINUS(x'''', 0)
IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))

The transformation is resulting in two new DP problems: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 6
                 ↳Forward Instantiation Transformation


Dependency Pairs:

IF(false, s(x''''), 0) -> MINUS(x'''', 0)
MINUS(s(x''), 0) -> IF(false, s(x''), 0)


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

IF(false, s(x''''), 0) -> MINUS(x'''', 0)
one new Dependency Pair is created:

IF(false, s(s(x''''')), 0) -> MINUS(s(x'''''), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 8
                 ↳Forward Instantiation Transformation


Dependency Pairs:

IF(false, s(s(x''''')), 0) -> MINUS(s(x'''''), 0)
MINUS(s(x''), 0) -> IF(false, s(x''), 0)


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

MINUS(s(x''), 0) -> IF(false, s(x''), 0)
one new Dependency Pair is created:

MINUS(s(s(x''''''')), 0) -> IF(false, s(s(x''''''')), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 10
                 ↳Argument Filtering and Ordering


Dependency Pairs:

MINUS(s(s(x''''''')), 0) -> IF(false, s(s(x''''''')), 0)
IF(false, s(s(x''''')), 0) -> MINUS(s(x'''''), 0)


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MINUS(s(s(x''''''')), 0) -> IF(false, s(s(x''''''')), 0)
IF(false, s(s(x''''')), 0) -> MINUS(s(x'''''), 0)


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:
{0, s, false} > {MINUS, IF}

resulting in one new DP problem.
Used Argument Filtering System:
IF(x1, x2, x3) -> IF(x1, x2, x3)
MINUS(x1, x2) -> MINUS(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 12
                 ↳Dependency Graph


Dependency Pair:


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pairs:

IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))
one new Dependency Pair is created:

IF(false, s(s(x''''')), s(y''''')) -> MINUS(s(x'''''), s(y'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation


Dependency Pairs:

IF(false, s(s(x''''')), s(y''''')) -> MINUS(s(x'''''), s(y'''''))
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




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

MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))
one new Dependency Pair is created:

MINUS(s(s(x''''''')), s(y''')) -> IF(le(s(x'''''''), y'''), s(s(x''''''')), s(y'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 11
                 ↳Argument Filtering and Ordering


Dependency Pairs:

MINUS(s(s(x''''''')), s(y''')) -> IF(le(s(x'''''''), y'''), s(s(x''''''')), s(y'''))
IF(false, s(s(x''''')), s(y''''')) -> MINUS(s(x'''''), s(y'''''))


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

IF(false, s(s(x''''')), s(y''''')) -> MINUS(s(x'''''), s(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:
IF(x1, x2, x3) -> x2
s(x1) -> s(x1)
MINUS(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 13
                 ↳Dependency Graph


Dependency Pair:

MINUS(s(s(x''''''')), s(y''')) -> IF(le(s(x'''''''), y'''), s(s(x''''''')), s(y'''))


Rules:


p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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