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) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
   if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
   if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
   if_reach_2(true, x, y, edge(u, v, i), h) -> true
   if_reach_2(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.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   IF_REACH_2(false, x, y, edge(u, v, i), h) -> OR(reach(x, y, i, h), reach(v, y, union(i, h), empty))
   IF_REACH_2(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)
   IF_REACH_2(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
   IF_REACH_2(false, x, y, edge(u, v, i), h) -> UNION(i, h)
   REACH(x, y, edge(u, v, i), h) -> IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
   REACH(x, y, edge(u, v, i), h) -> EQ(x, u)
   UNION(edge(x, y, i), h) -> UNION(i, h)
   IF_REACH_1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
   IF_REACH_1(true, x, y, edge(u, v, i), h) -> IF_REACH_2(eq(y, v), x, y, edge(u, v, i), h)
   IF_REACH_1(true, x, y, edge(u, v, i), h) -> EQ(y, v)
   EQ(s(x), s(y)) -> EQ(x, y)

Furthermore, R contains three SCCs.


SCC1:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(EQ(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(edge(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
POL(UNION(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

IF_REACH_2(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
IF_REACH_1(true, x, y, edge(u, v, i), h) -> IF_REACH_2(eq(y, v), x, y, edge(u, v, i), h)
IF_REACH_1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
REACH(x, y, edge(u, v, i), h) -> IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
IF_REACH_2(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

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


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(IF_REACH_2(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5
POL(s(x_1)) = 0
POL(eq(x_1, x_2)) = 0
POL(true) = 0
POL(edge(x_1, x_2, x_3)) = 1 + x_3
POL(union(x_1, x_2)) = x_1 + x_2
POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5
POL(empty) = 0
POL(0) = 0
POL(REACH(x_1, x_2, x_3, x_4)) = x_3 + x_4
POL(false) = 0

 resulting in one subcycle.


SCC3.Polo1:

REACH(x, y, edge(u, v, i), h) -> IF_REACH_1(eq(x, u), x, y, edge(u, v, i), h)
IF_REACH_1(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 0
POL(eq(x_1, x_2)) = 0
POL(true) = 0
POL(edge(x_1, x_2, x_3)) = 1 + x_3
POL(IF_REACH_1(x_1, x_2, x_3, x_4, x_5)) = x_4
POL(0) = 0
POL(REACH(x_1, x_2, x_3, x_4)) = 1 + x_3
POL(false) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.544 seconds.