Term Rewriting System R:
[x, y, h, i, u, v]
eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

EQ(s(x), s(y)) -> EQ(x, y)
UNION(edge(x, y, i), h) -> UNION(i, h)
REACH(x, y, edge(u, v, i), h) -> IFREACH1(eq(x, u), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> EQ(x, u)
IFREACH1(true, x, y, edge(u, v, i), h) -> IFREACH2(eq(y, v), x, y, edge(u, v, i), h)
IFREACH1(true, x, y, edge(u, v, i), h) -> EQ(y, v)
IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
IFREACH2(false, x, y, edge(u, v, i), h) -> OR(reach(x, y, i, h), reach(v, y, union(i, h), empty))
IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)
IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
IFREACH2(false, x, y, edge(u, v, i), h) -> UNION(i, h)

Furthermore, R contains three SCCs. 


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


Dependency Pair:

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


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

UNION(edge(x, y, i), h) -> UNION(i, h)


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





The following dependency pair can be strictly oriented:

UNION(edge(x, y, i), h) -> UNION(i, h)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(edge(x1, x2, x3))=  1 + x3  
  POL(UNION(x1, x2))=  x1  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Narrowing Transformation


Dependency Pairs:

IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)
IFREACH1(true, x, y, edge(u, v, i), h) -> IFREACH2(eq(y, v), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> IFREACH1(eq(x, u), x, y, edge(u, v, i), h)


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





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

IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
two new Dependency Pairs are created:

IFREACH2(false, x, y, edge(u, v, empty), h'') -> REACH(v, y, h'', empty)
IFREACH2(false, x, y, edge(u, v, edge(x'', y'', i'')), h'') -> REACH(v, y, edge(x'', y'', union(i'', h'')), empty)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Polynomial Ordering


Dependency Pairs:

IFREACH2(false, x, y, edge(u, v, edge(x'', y'', i'')), h'') -> REACH(v, y, edge(x'', y'', union(i'', h'')), empty)
IFREACH2(false, x, y, edge(u, v, empty), h'') -> REACH(v, y, h'', empty)
IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)
IFREACH1(true, x, y, edge(u, v, i), h) -> IFREACH2(eq(y, v), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> IFREACH1(eq(x, u), x, y, edge(u, v, i), h)
IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





The following dependency pairs can be strictly oriented:

IFREACH2(false, x, y, edge(u, v, edge(x'', y'', i'')), h'') -> REACH(v, y, edge(x'', y'', union(i'', h'')), empty)
IFREACH2(false, x, y, edge(u, v, empty), h'') -> REACH(v, y, h'', empty)
IFREACH2(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(edge(x1, x2, x3))=  1 + x3  
  POL(eq(x1, x2))=  0  
  POL(0)=  0  
  POL(false)=  0  
  POL(union(x1, x2))=  x1 + x2  
  POL(IF_REACH_2(x1, x2, x3, x4, x5))=  x4 + x5  
  POL(REACH(x1, x2, x3, x4))=  x3 + x4  
  POL(true)=  0  
  POL(s(x1))=  0  
  POL(empty)=  0  
  POL(IF_REACH_1(x1, x2, x3, x4, x5))=  x4 + x5  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Polo
             ...
               →DP Problem 7
                 ↳Dependency Graph


Dependency Pairs:

IFREACH1(true, x, y, edge(u, v, i), h) -> IFREACH2(eq(y, v), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> IFREACH1(eq(x, u), x, y, edge(u, v, i), h)
IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Polo
             ...
               →DP Problem 8
                 ↳Instantiation Transformation


Dependency Pairs:

IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
REACH(x, y, edge(u, v, i), h) -> IFREACH1(eq(x, u), x, y, edge(u, v, i), h)


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





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

REACH(x, y, edge(u, v, i), h) -> IFREACH1(eq(x, u), x, y, edge(u, v, i), h)
one new Dependency Pair is created:

REACH(x'', y'', edge(u'', v0, i''), edge(u''', v'', h'')) -> IFREACH1(eq(x'', u''), x'', y'', edge(u'', v0, i''), edge(u''', v'', h''))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

REACH(x'', y'', edge(u'', v0, i''), edge(u''', v'', h'')) -> IFREACH1(eq(x'', u''), x'', y'', edge(u'', v0, i''), edge(u''', v'', h''))
IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





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

IFREACH1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
one new Dependency Pair is created:

IFREACH1(false, x', y', edge(u', v', i'), edge(u'''''', v'''', h'''')) -> REACH(x', y', i', edge(u', v', edge(u'''''', v'''', h'''')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

IFREACH1(false, x', y', edge(u', v', i'), edge(u'''''', v'''', h'''')) -> REACH(x', y', i', edge(u', v', edge(u'''''', v'''', h'''')))
REACH(x'', y'', edge(u'', v0, i''), edge(u''', v'', h'')) -> IFREACH1(eq(x'', u''), x'', y'', edge(u'', v0, i''), edge(u''', v'', h''))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





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

REACH(x'', y'', edge(u'', v0, i''), edge(u''', v'', h'')) -> IFREACH1(eq(x'', u''), x'', y'', edge(u'', v0, i''), edge(u''', v'', h''))
one new Dependency Pair is created:

REACH(x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h''''''))) -> IFREACH1(eq(x'''', u''0), x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h'''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Polo
             ...
               →DP Problem 11
                 ↳Instantiation Transformation


Dependency Pairs:

REACH(x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h''''''))) -> IFREACH1(eq(x'''', u''0), x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h'''''')))
IFREACH1(false, x', y', edge(u', v', i'), edge(u'''''', v'''', h'''')) -> REACH(x', y', i', edge(u', v', edge(u'''''', v'''', h'''')))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





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

IFREACH1(false, x', y', edge(u', v', i'), edge(u'''''', v'''', h'''')) -> REACH(x', y', i', edge(u', v', edge(u'''''', v'''', h'''')))
one new Dependency Pair is created:

IFREACH1(false, x''', y'', edge(u'', v'', i''), edge(u''''''', v'''''', edge(u'''''''''', v'''''''', h''''''''))) -> REACH(x''', y'', i'', edge(u'', v'', edge(u''''''', v'''''', edge(u'''''''''', v'''''''', h''''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Polo
             ...
               →DP Problem 12
                 ↳Polynomial Ordering


Dependency Pairs:

IFREACH1(false, x''', y'', edge(u'', v'', i''), edge(u''''''', v'''''', edge(u'''''''''', v'''''''', h''''''''))) -> REACH(x''', y'', i'', edge(u'', v'', edge(u''''''', v'''''', edge(u'''''''''', v'''''''', h''''''''))))
REACH(x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h''''''))) -> IFREACH1(eq(x'''', u''0), x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h'''''')))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





The following dependency pair can be strictly oriented:

IFREACH1(false, x''', y'', edge(u'', v'', i''), edge(u''''''', v'''''', edge(u'''''''''', v'''''''', h''''''''))) -> REACH(x''', y'', i'', edge(u'', v'', edge(u''''''', v'''''', edge(u'''''''''', v'''''''', h''''''''))))


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(edge(x1, x2, x3))=  1 + x3  
  POL(eq(x1, x2))=  0  
  POL(0)=  0  
  POL(false)=  0  
  POL(REACH(x1, x2, x3, x4))=  x3  
  POL(true)=  0  
  POL(s(x1))=  0  
  POL(IF_REACH_1(x1, x2, x3, x4, x5))=  x4  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 6
             ↳Polo
             ...
               →DP Problem 13
                 ↳Dependency Graph


Dependency Pair:

REACH(x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h''''''))) -> IFREACH1(eq(x'''', u''0), x'''', y'''', edge(u''0, v0', i''''), edge(u'''', v'''', edge(u'''''''', v'''''', h'''''')))


Rules:


eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> ifreach1(eq(x, u), x, y, edge(u, v, i), h)
ifreach1(true, x, y, edge(u, v, i), h) -> ifreach2(eq(y, v), x, y, edge(u, v, i), h)
ifreach1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
ifreach2(true, x, y, edge(u, v, i), h) -> true
ifreach2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))





Using the Dependency Graph resulted in no new DP problems.

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