Termination Proof using AProVE

Term Rewriting System R:
[x]
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

REC(rec(x)) -> SENT(rec(x))
REC(sent(x)) -> SENT(rec(x))
REC(sent(x)) -> REC(x)
REC(no(x)) -> SENT(rec(x))
REC(no(x)) -> REC(x)
REC(bot) -> SENT(bot)
REC(up(x)) -> REC(x)
SENT(up(x)) -> SENT(x)
NO(up(x)) -> NO(x)
TOP(rec(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> CHECK(rec(x))
TOP(rec(up(x))) -> REC(x)
TOP(sent(up(x))) -> TOP(check(rec(x)))
TOP(sent(up(x))) -> CHECK(rec(x))
TOP(sent(up(x))) -> REC(x)
TOP(no(up(x))) -> TOP(check(rec(x)))
TOP(no(up(x))) -> CHECK(rec(x))
TOP(no(up(x))) -> REC(x)
CHECK(up(x)) -> CHECK(x)
CHECK(sent(x)) -> SENT(check(x))
CHECK(sent(x)) -> CHECK(x)
CHECK(rec(x)) -> REC(check(x))
CHECK(rec(x)) -> CHECK(x)
CHECK(no(x)) -> NO(check(x))
CHECK(no(x)) -> CHECK(x)

Furthermore, R contains six SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

SENT(up(x)) -> SENT(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

SENT(up(x)) -> SENT(x)

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(up(x₁)) = 1 + x₁

POL(SENT(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

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

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

NO(up(x)) -> NO(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

NO(up(x)) -> NO(x)

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(NO(x₁)) = x₁

POL(up(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 8

             ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

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

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

REC(up(x)) -> REC(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

REC(up(x)) -> REC(x)

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(up(x₁)) = 1 + x₁

POL(REC(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 9

             ↳Dependency Graph

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pairs:

REC(no(x)) -> REC(x)
REC(sent(x)) -> REC(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

REC(no(x)) -> REC(x)

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(no(x₁)) = 1 + x₁

POL(REC(x₁)) = x₁

POL(sent(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 10

             ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

REC(sent(x)) -> REC(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

REC(sent(x)) -> REC(x)

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(REC(x₁)) = x₁

POL(sent(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 10

             ↳Polo

...

               →DP Problem 11

                 ↳Dependency Graph

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

Dependency Pair:

CHECK(up(x)) -> CHECK(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

CHECK(up(x)) -> CHECK(x)

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(up(x₁)) = 1 + x₁

POL(CHECK(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 12

             ↳Dependency Graph

       →DP Problem 6

         ↳Polo

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polynomial Ordering

Dependency Pairs:

CHECK(no(x)) -> CHECK(x)
CHECK(sent(x)) -> CHECK(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

CHECK(no(x)) -> CHECK(x)

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(no(x₁)) = 1 + x₁

POL(CHECK(x₁)) = x₁

POL(sent(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

           →DP Problem 13

             ↳Polynomial Ordering

Dependency Pair:

CHECK(sent(x)) -> CHECK(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

CHECK(sent(x)) -> CHECK(x)

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(CHECK(x₁)) = x₁

POL(sent(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

           →DP Problem 13

             ↳Polo

...

               →DP Problem 14

                 ↳Dependency Graph

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

POL(no(x₁))	= 1 + x₁
POL(REC(x₁))	= x₁
POL(sent(x₁))	= x₁

POL(no(x₁))	= 1 + x₁
POL(CHECK(x₁))	= x₁
POL(sent(x₁))	= x₁