Termination w.r.t. Q proof of /tmp/tmpI55QmY/10.trs

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 Narrowing (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 QDP

↳9 Narrowing (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 Narrowing (⇔)

↳16 QDP

↳17 UsableRulesProof (⇔)

↳18 QDP

↳19 MNOCProof (⇔)

↳20 QDP

↳21 Rewriting (⇔)

↳22 QDP

↳23 DependencyGraphProof (⇔)

↳24 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(c(c(y))) → C(c(a(y, 0)))
C(c(c(y))) → C(a(y, 0))
C(a(a(0, x), y)) → C(c(c(0)))
C(a(a(0, x), y)) → C(c(0))
C(a(a(0, x), y)) → C(0)

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(c(c(y))) → C(a(y, 0))
C(a(a(0, x), y)) → C(c(c(0)))
C(c(c(y))) → C(c(a(y, 0)))
C(a(a(0, x), y)) → C(c(0))

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule C(a(a(0, x), y)) → C(c(0)) at position [0] we obtained the following new rules [LPAR04]:

C(a(a(0, y0), y1)) → C(0)

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(c(c(y))) → C(a(y, 0))
C(a(a(0, x), y)) → C(c(c(0)))
C(c(c(y))) → C(c(a(y, 0)))
C(a(a(0, y0), y1)) → C(0)

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, x), y)) → C(c(c(0)))
C(c(c(y))) → C(c(a(y, 0)))
C(c(c(y))) → C(a(y, 0))

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule C(c(c(y))) → C(c(a(y, 0))) at position [0] we obtained the following new rules [LPAR04]:

C(c(c(a(0, x0)))) → C(a(c(c(c(0))), 0))
C(c(c(y0))) → C(a(y0, 0))

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, x), y)) → C(c(c(0)))
C(c(c(y))) → C(a(y, 0))
C(c(c(a(0, x0)))) → C(a(c(c(c(0))), 0))

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

C(c(c(a(0, x0)))) → C(a(c(c(c(0))), 0))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(C(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(a(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(0) =
/ 0 \

\ 0 /

POL(c(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁

The following usable rules [FROCOS05] were oriented:

c(y) → y
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(c(c(y))) → c(c(a(y, 0)))

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, x), y)) → C(c(c(0)))
C(c(c(y))) → C(a(y, 0))

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

C(c(c(y))) → C(a(y, 0))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(C(x₁)) =
/ -I \

| -I |

\ 0A /

+
/ 0A -I -I \

| -I -I -I |

\ 3A 0A 2A /

· x₁

POL(a(x₁, x₂)) =
/ -I \

| 0A |

\ 2A /

+
/ 0A -I 0A \

| 0A -I -I |

\ -I -I -I /

· x₁ +
/ -I -I 0A \

| -I 0A -I |

\ -I -I -I /

· x₂

POL(0) =
/ 0A \

| 0A |

\ -I /

POL(c(x₁)) =
/ 0A \

| 2A |

\ -I /

+
/ 1A 0A -I \

| -I 1A 1A |

\ 0A -I 0A /

· x₁

The following usable rules [FROCOS05] were oriented:

c(y) → y
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(c(c(y))) → c(c(a(y, 0)))

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, x), y)) → C(c(c(0)))

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule C(a(a(0, x), y)) → C(c(c(0))) at position [0] we obtained the following new rules [LPAR04]:

C(a(a(0, y0), y1)) → C(c(0))

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, y0), y1)) → C(c(0))

The TRS R consists of the following rules:

c(c(c(y))) → c(c(a(y, 0)))
c(a(a(0, x), y)) → a(c(c(c(0))), y)
c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, y0), y1)) → C(c(0))

The TRS R consists of the following rules:

c(y) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, y0), y1)) → C(c(0))

The TRS R consists of the following rules:

c(y) → y

The set Q consists of the following terms:

c(x0)

We have to consider all minimal (P,Q,R)-chains.

(21) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule C(a(a(0, y0), y1)) → C(c(0)) at position [0] we obtained the following new rules [LPAR04]:

C(a(a(0, y0), y1)) → C(0)

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(a(a(0, y0), y1)) → C(0)

The TRS R consists of the following rules:

c(y) → y

The set Q consists of the following terms:

c(x0)

We have to consider all minimal (P,Q,R)-chains.

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) Narrowing (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) Narrowing (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) Narrowing (EQUIVALENT transformation)

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) MNOCProof (EQUIVALENT transformation)

(20) Obligation:

(21) Rewriting (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) TRUE