Termination Proof using AProVE

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_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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.

     ↳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) -> IF_REACH₁(eq(x, u), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> EQ(x, u)
IF_REACH₁(true, x, y, edge(u, v, i), h) -> IF_REACH₂(eq(y, v), x, y, edge(u, v, i), h)
IF_REACH₁(true, x, y, edge(u, v, i), h) -> EQ(y, v)
IF_REACH₁(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
IF_REACH₂(false, x, y, edge(u, v, i), h) -> OR(reach(x, y, i, h), reach(v, y, union(i, h), empty))
IF_REACH₂(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)
IF_REACH₂(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
IF_REACH₂(false, x, y, edge(u, v, i), h) -> UNION(i, h)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳Polo

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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

Dependency Pairs:

IF_REACH₁(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
IF_REACH₂(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
IF_REACH₂(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)
IF_REACH₁(true, x, y, edge(u, v, i), h) -> IF_REACH₂(eq(y, v), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> IF_REACH₁(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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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:

IF_REACH₂(false, x, y, edge(u, v, i), h) -> REACH(v, y, union(i, h), empty)
IF_REACH₂(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, h)

Additionally, the following usable rules for innermost 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳Dependency Graph

Dependency Pairs:

IF_REACH₁(false, x, y, edge(u, v, i), h) -> REACH(x, y, i, edge(u, v, h))
IF_REACH₁(true, x, y, edge(u, v, i), h) -> IF_REACH₂(eq(y, v), x, y, edge(u, v, i), h)
REACH(x, y, edge(u, v, i), h) -> IF_REACH₁(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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳DGraph

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

REACH(x, y, edge(u, v, i), h) -> IF_REACH₁(eq(x, u), x, y, edge(u, v, i), h)
IF_REACH₁(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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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:

IF_REACH₁(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 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳DGraph

...

               →DP Problem 8

                 ↳Dependency Graph

Dependency Pair:

REACH(x, y, edge(u, v, i), h) -> IF_REACH₁(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) -> if_reach₁(eq(x, u), x, y, edge(u, v, i), h)
if_reach₁(true, x, y, edge(u, v, i), h) -> if_reach₂(eq(y, v), x, y, edge(u, v, i), h)
if_reach₁(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))
if_reach₂(true, x, y, edge(u, v, i), h) -> true
if_reach₂(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:
0:00 minutes

POL(edge(x₁, x₂, x₃))	= 1 + x₃
POL(UNION(x₁, x₂))	= x₁

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