Termination w.r.t. Q proof of /tmp/tmpWi9FTb/LengthOfFiniteLists_nosorts_noand

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

0 QTRS

↳1 QTRSRRRProof (⇔, 80 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 15 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 38 ms)

↳6 QTRS

↳7 QTRSRRRProof (⇔, 0 ms)

↳8 QTRS

↳9 QTRSRRRProof (⇔, 0 ms)

↳10 QTRS

↳11 QTRSRRRProof (⇔, 11 ms)

↳12 QTRS

↳13 DependencyPairsProof (⇔, 0 ms)

↳14 QDP

↳15 DependencyGraphProof (⇔, 0 ms)

↳16 AND

    ↳17 QDP

        ↳18 QDPSizeChangeProof (⇔, 0 ms)

        ↳19 YES

    ↳20 QDP

        ↳21 NonLoopProof (⇔, 805 ms)

        ↳22 NO

(0) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(U12(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = x₁ + 2·x₂
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__length(nil) → 0

(2) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U12(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(a__U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__length(x₁)) = 1 + x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = 2·x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))

(4) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__length(x₁)) = 1 + 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)

(6) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(a__U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__length(x₁)) = x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(mark(x₁)) = 2·x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(nil) → nil

(8) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__zeros → zeros

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(a__U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__length(x₁)) = x₁
POL(a__zeros) = 2
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(mark(x₁)) = 2·x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__zeros → zeros

(10) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(a__U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__length(x₁)) = x₁
POL(a__zeros) = 1
POL(cons(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(mark(x₁)) = 1 + 2·x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt

(12) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))

Q is empty.

(13) DependencyPairsProof (EQUIVALENT transformation)

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

(14) Obligation:

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

A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__U12(tt, L) → MARK(L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.

(16) Complex Obligation (AND)

(17) Obligation:

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

MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))

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

(18) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1

(19) YES

(20) Obligation:

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

A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))

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

(21) NonLoopProof (EQUIVALENT transformation)

By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [ ] on the rule
A__U11(tt, zeros)[ ]ⁿ[ ] → A__U11(tt, zeros)[ ]ⁿ[ ]
This rule is correct for the QDP as the following derivation shows:

A__U11(tt, zeros)[ ]ⁿ[ ] → A__U11(tt, zeros)[ ]ⁿ[ ]
    by Narrowing at position: []
        intermediate steps: Instantiation - Instantiation
        A__U11(tt, L)[ ]ⁿ[ ] → A__U12(tt, L)[ ]ⁿ[ ]
            by OriginalRule from TRS P

        A__U12(tt, zeros)[ ]ⁿ[ ] → A__U11(tt, zeros)[ ]ⁿ[ ]
            by Narrowing at position: []
                A__U12(tt, zeros)[ ]ⁿ[ ] → A__LENGTH(cons(0, zeros))[ ]ⁿ[ ]
                    by Narrowing at position: [0]
                        A__U12(tt, zeros)[ ]ⁿ[ ] → A__LENGTH(a__zeros)[ ]ⁿ[ ]
                            by Narrowing at position: [0]
                                intermediate steps: Instantiation - Instantiation
                                A__U12(tt, L)[ ]ⁿ[ ] → A__LENGTH(mark(L))[ ]ⁿ[ ]
                                    by OriginalRule from TRS P

                                mark(zeros)[ ]ⁿ[ ] → a__zeros[ ]ⁿ[ ]
                                    by OriginalRule from TRS R

                        a__zeros[ ]ⁿ[ ] → cons(0, zeros)[ ]ⁿ[ ]
                            by OriginalRule from TRS R

                intermediate steps: Instantiation - Instantiation - Instantiation
                A__LENGTH(cons(N, L))[ ]ⁿ[ ] → A__U11(tt, L)[ ]ⁿ[ ]
                    by OriginalRule from TRS P

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) QTRSRRRProof (EQUIVALENT transformation)

(10) Obligation:

(11) QTRSRRRProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyPairsProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) Complex Obligation (AND)

(17) Obligation:

(18) QDPSizeChangeProof (EQUIVALENT transformation)

(19) YES

(20) Obligation:

(21) NonLoopProof (EQUIVALENT transformation)

(22) NO