Term Rewriting System R:
[x, y]
f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x)) -> F(x)
G(s(x), s(y)) -> IF(f(x), s(x), s(y))
G(s(x), s(y)) -> F(x)
G(x, c(y)) -> G(x, g(s(c(y)), y))
G(x, c(y)) -> G(s(c(y)), y)

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation
       →DP Problem 2
Nar


Dependency Pair:

F(s(x)) -> F(x)


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

F(s(x)) -> F(x)
one new Dependency Pair is created:

F(s(s(x''))) -> F(s(x''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
Forward Instantiation Transformation
       →DP Problem 2
Nar


Dependency Pair:

F(s(s(x''))) -> F(s(x''))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

F(s(s(x''))) -> F(s(x''))
one new Dependency Pair is created:

F(s(s(s(x'''')))) -> F(s(s(x'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
FwdInst
             ...
               →DP Problem 4
Argument Filtering and Ordering
       →DP Problem 2
Nar


Dependency Pair:

F(s(s(s(x'''')))) -> F(s(s(x'''')))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(s(s(s(x'''')))) -> F(s(s(x'''')))


There are no usable rules for innermost 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:
F(x1) -> F(x1)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Narrowing Transformation


Dependency Pairs:

G(x, c(y)) -> G(s(c(y)), y)
G(x, c(y)) -> G(x, g(s(c(y)), y))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

G(x, c(y)) -> G(x, g(s(c(y)), y))
two new Dependency Pairs are created:

G(x, c(s(y''))) -> G(x, if(f(c(s(y''))), s(c(s(y''))), s(y'')))
G(x, c(c(y''))) -> G(x, g(s(c(c(y''))), g(s(c(y'')), y'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Narrowing Transformation


Dependency Pairs:

G(x, c(c(y''))) -> G(x, g(s(c(c(y''))), g(s(c(y'')), y'')))
G(x, c(s(y''))) -> G(x, if(f(c(s(y''))), s(c(s(y''))), s(y'')))
G(x, c(y)) -> G(s(c(y)), y)


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

G(x, c(s(y''))) -> G(x, if(f(c(s(y''))), s(c(s(y''))), s(y'')))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 7
Narrowing Transformation


Dependency Pairs:

G(x, c(y)) -> G(s(c(y)), y)
G(x, c(c(y''))) -> G(x, g(s(c(c(y''))), g(s(c(y'')), y'')))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

G(x, c(c(y''))) -> G(x, g(s(c(c(y''))), g(s(c(y'')), y'')))
two new Dependency Pairs are created:

G(x, c(c(s(y')))) -> G(x, g(s(c(c(s(y')))), if(f(c(s(y'))), s(c(s(y'))), s(y'))))
G(x, c(c(c(y')))) -> G(x, g(s(c(c(c(y')))), g(s(c(c(y'))), g(s(c(y')), y'))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 8
Narrowing Transformation


Dependency Pairs:

G(x, c(c(c(y')))) -> G(x, g(s(c(c(c(y')))), g(s(c(c(y'))), g(s(c(y')), y'))))
G(x, c(c(s(y')))) -> G(x, g(s(c(c(s(y')))), if(f(c(s(y'))), s(c(s(y'))), s(y'))))
G(x, c(y)) -> G(s(c(y)), y)


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

G(x, c(c(s(y')))) -> G(x, g(s(c(c(s(y')))), if(f(c(s(y'))), s(c(s(y'))), s(y'))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 9
Forward Instantiation Transformation


Dependency Pairs:

G(x, c(y)) -> G(s(c(y)), y)
G(x, c(c(c(y')))) -> G(x, g(s(c(c(c(y')))), g(s(c(c(y'))), g(s(c(y')), y'))))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

G(x, c(y)) -> G(s(c(y)), y)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 10
Forward Instantiation Transformation


Dependency Pairs:

G(x, c(c(c(c(y'''))))) -> G(s(c(c(c(c(y'''))))), c(c(c(y'''))))
G(x, c(c(y''))) -> G(s(c(c(y''))), c(y''))
G(x, c(c(c(y')))) -> G(x, g(s(c(c(c(y')))), g(s(c(c(y'))), g(s(c(y')), y'))))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




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

G(x, c(c(y''))) -> G(s(c(c(y''))), c(y''))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Nar
             ...
               →DP Problem 11
Argument Filtering and Ordering


Dependency Pairs:

G(x, c(c(c(c(c(y''''')))))) -> G(s(c(c(c(c(c(y''''')))))), c(c(c(c(y''''')))))
G(x, c(c(c(y'''')))) -> G(s(c(c(c(y'''')))), c(c(y'''')))
G(x, c(c(c(c(y''''))))) -> G(s(c(c(c(c(y''''))))), c(c(c(y''''))))
G(x, c(c(c(y')))) -> G(x, g(s(c(c(c(y')))), g(s(c(c(y'))), g(s(c(y')), y'))))
G(x, c(c(c(c(y'''))))) -> G(s(c(c(c(c(y'''))))), c(c(c(y'''))))


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

G(x, c(c(c(c(c(y''''')))))) -> G(s(c(c(c(c(c(y''''')))))), c(c(c(c(y''''')))))
G(x, c(c(c(y'''')))) -> G(s(c(c(c(y'''')))), c(c(y'''')))
G(x, c(c(c(c(y''''))))) -> G(s(c(c(c(c(y''''))))), c(c(c(y''''))))
G(x, c(c(c(y')))) -> G(x, g(s(c(c(c(y')))), g(s(c(c(y'))), g(s(c(y')), y'))))
G(x, c(c(c(c(y'''))))) -> G(s(c(c(c(c(y'''))))), c(c(c(y'''))))


The following usable rules for innermost can be oriented:

g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))
if(true, x, y) -> x
if(false, x, y) -> y
f(0) -> true
f(1) -> false
f(s(x)) -> f(x)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{c, f, G, s} > true
{c, f, G, s} > g > if
{c, f, G, s} > false

resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> G(x1, x2)
c(x1) -> c(x1)
g(x1, x2) -> g(x1, x2)
s(x1) -> s
if(x1, x2, x3) -> if(x1, x2, x3)
f(x1) -> f


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


Dependency Pair:


Rules:


f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes