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))

Innermost 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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(EQ(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 3`
`         ↳AFS`

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(edge(x1, x2, x3)) =  1 + x1 + x2 + x3 POL(UNION(x1, x2)) =  x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`

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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`

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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
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 + x1 + x2 + x3 POL(eq) =  0 POL(false) =  0 POL(IF_REACH_2(x1, x2, x3, x4, x5)) =  x1 + x2 + x3 + x4 + x5 POL(union(x1, x2)) =  x1 + x2 POL(REACH(x1, x2, x3, x4)) =  x1 + x2 + x3 + x4 POL(true) =  0 POL(empty) =  0 POL(IF_REACH_1(x1, x2, x3, x4, x5)) =  x1 + x2 + x3 + x4 + x5

resulting in one new DP problem.
Used Argument Filtering System:
IFREACH2(x1, x2, x3, x4, x5) -> IFREACH2(x1, x2, x3, x4, x5)
REACH(x1, x2, x3, x4) -> REACH(x1, x2, x3, x4)
edge(x1, x2, x3) -> edge(x1, x2, x3)
IFREACH1(x1, x2, x3, x4, x5) -> IFREACH1(x1, x2, x3, x4, x5)
eq(x1, x2) -> eq
union(x1, x2) -> union(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`

Dependency Pairs:

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

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳DGraph`
`             ...`
`               →DP Problem 7`
`                 ↳Argument Filtering and Ordering`

Dependency Pairs:

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(edge(x1, x2, x3)) =  1 + x1 + x2 + x3

resulting in one new DP problem.
Used Argument Filtering System:
REACH(x1, x2, x3, x4) -> x3
edge(x1, x2, x3) -> edge(x1, x2, x3)
IFREACH1(x1, x2, x3, x4, x5) -> x4

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳DGraph`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`

Dependency Pair:

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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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