Left Termination of the query pattern len1_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(0) Obligation:

Clauses:

len1([], 0).
len1(.(X1, Ts), N) :- ','(len1(Ts, M), eq(N, s(M))).
eq(X, X).

Queries:

len1(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

len120([], 0).
len120(.(T24, T25), X49) :- len120(T25, X48).
len120(.(T24, T25), s(T29)) :- len120(T25, T29).
len11([], 0).
len11(.(T6, []), s(0)).
len11(.(T6, .(T17, T18)), T9) :- len120(T18, X31).
len11(.(T6, .(T17, T18)), s(s(T38))) :- len120(T18, T38).

Queries:

len11(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
len11_in: (b,f)
len120_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

len11_in_ga([], 0) → len11_out_ga([], 0)
len11_in_ga(.(T6, []), s(0)) → len11_out_ga(.(T6, []), s(0))
len11_in_ga(.(T6, .(T17, T18)), T9) → U3_ga(T6, T17, T18, T9, len120_in_ga(T18, X31))
len120_in_ga([], 0) → len120_out_ga([], 0)
len120_in_ga(.(T24, T25), X49) → U1_ga(T24, T25, X49, len120_in_ga(T25, X48))
len120_in_ga(.(T24, T25), s(T29)) → U2_ga(T24, T25, T29, len120_in_ga(T25, T29))
U2_ga(T24, T25, T29, len120_out_ga(T25, T29)) → len120_out_ga(.(T24, T25), s(T29))
U1_ga(T24, T25, X49, len120_out_ga(T25, X48)) → len120_out_ga(.(T24, T25), X49)
U3_ga(T6, T17, T18, T9, len120_out_ga(T18, X31)) → len11_out_ga(.(T6, .(T17, T18)), T9)
len11_in_ga(.(T6, .(T17, T18)), s(s(T38))) → U4_ga(T6, T17, T18, T38, len120_in_ga(T18, T38))
U4_ga(T6, T17, T18, T38, len120_out_ga(T18, T38)) → len11_out_ga(.(T6, .(T17, T18)), s(s(T38)))

The argument filtering Pi contains the following mapping:
len11_in_ga(x1, x2)  =  len11_in_ga(x1)
[]  =  []
len11_out_ga(x1, x2)  =  len11_out_ga
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
len120_out_ga(x1, x2)  =  len120_out_ga
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

len11_in_ga([], 0) → len11_out_ga([], 0)
len11_in_ga(.(T6, []), s(0)) → len11_out_ga(.(T6, []), s(0))
len11_in_ga(.(T6, .(T17, T18)), T9) → U3_ga(T6, T17, T18, T9, len120_in_ga(T18, X31))
len120_in_ga([], 0) → len120_out_ga([], 0)
len120_in_ga(.(T24, T25), X49) → U1_ga(T24, T25, X49, len120_in_ga(T25, X48))
len120_in_ga(.(T24, T25), s(T29)) → U2_ga(T24, T25, T29, len120_in_ga(T25, T29))
U2_ga(T24, T25, T29, len120_out_ga(T25, T29)) → len120_out_ga(.(T24, T25), s(T29))
U1_ga(T24, T25, X49, len120_out_ga(T25, X48)) → len120_out_ga(.(T24, T25), X49)
U3_ga(T6, T17, T18, T9, len120_out_ga(T18, X31)) → len11_out_ga(.(T6, .(T17, T18)), T9)
len11_in_ga(.(T6, .(T17, T18)), s(s(T38))) → U4_ga(T6, T17, T18, T38, len120_in_ga(T18, T38))
U4_ga(T6, T17, T18, T38, len120_out_ga(T18, T38)) → len11_out_ga(.(T6, .(T17, T18)), s(s(T38)))

The argument filtering Pi contains the following mapping:
len11_in_ga(x1, x2)  =  len11_in_ga(x1)
[]  =  []
len11_out_ga(x1, x2)  =  len11_out_ga
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
len120_out_ga(x1, x2)  =  len120_out_ga
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

LEN11_IN_GA(.(T6, .(T17, T18)), T9) → U3_GA(T6, T17, T18, T9, len120_in_ga(T18, X31))
LEN11_IN_GA(.(T6, .(T17, T18)), T9) → LEN120_IN_GA(T18, X31)
LEN120_IN_GA(.(T24, T25), X49) → U1_GA(T24, T25, X49, len120_in_ga(T25, X48))
LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)
LEN120_IN_GA(.(T24, T25), s(T29)) → U2_GA(T24, T25, T29, len120_in_ga(T25, T29))
LEN120_IN_GA(.(T24, T25), s(T29)) → LEN120_IN_GA(T25, T29)
LEN11_IN_GA(.(T6, .(T17, T18)), s(s(T38))) → U4_GA(T6, T17, T18, T38, len120_in_ga(T18, T38))
LEN11_IN_GA(.(T6, .(T17, T18)), s(s(T38))) → LEN120_IN_GA(T18, T38)

The TRS R consists of the following rules:

len11_in_ga([], 0) → len11_out_ga([], 0)
len11_in_ga(.(T6, []), s(0)) → len11_out_ga(.(T6, []), s(0))
len11_in_ga(.(T6, .(T17, T18)), T9) → U3_ga(T6, T17, T18, T9, len120_in_ga(T18, X31))
len120_in_ga([], 0) → len120_out_ga([], 0)
len120_in_ga(.(T24, T25), X49) → U1_ga(T24, T25, X49, len120_in_ga(T25, X48))
len120_in_ga(.(T24, T25), s(T29)) → U2_ga(T24, T25, T29, len120_in_ga(T25, T29))
U2_ga(T24, T25, T29, len120_out_ga(T25, T29)) → len120_out_ga(.(T24, T25), s(T29))
U1_ga(T24, T25, X49, len120_out_ga(T25, X48)) → len120_out_ga(.(T24, T25), X49)
U3_ga(T6, T17, T18, T9, len120_out_ga(T18, X31)) → len11_out_ga(.(T6, .(T17, T18)), T9)
len11_in_ga(.(T6, .(T17, T18)), s(s(T38))) → U4_ga(T6, T17, T18, T38, len120_in_ga(T18, T38))
U4_ga(T6, T17, T18, T38, len120_out_ga(T18, T38)) → len11_out_ga(.(T6, .(T17, T18)), s(s(T38)))

The argument filtering Pi contains the following mapping:
len11_in_ga(x1, x2)  =  len11_in_ga(x1)
[]  =  []
len11_out_ga(x1, x2)  =  len11_out_ga
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
len120_out_ga(x1, x2)  =  len120_out_ga
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
LEN11_IN_GA(x1, x2)  =  LEN11_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

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

LEN11_IN_GA(.(T6, .(T17, T18)), T9) → U3_GA(T6, T17, T18, T9, len120_in_ga(T18, X31))
LEN11_IN_GA(.(T6, .(T17, T18)), T9) → LEN120_IN_GA(T18, X31)
LEN120_IN_GA(.(T24, T25), X49) → U1_GA(T24, T25, X49, len120_in_ga(T25, X48))
LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)
LEN120_IN_GA(.(T24, T25), s(T29)) → U2_GA(T24, T25, T29, len120_in_ga(T25, T29))
LEN120_IN_GA(.(T24, T25), s(T29)) → LEN120_IN_GA(T25, T29)
LEN11_IN_GA(.(T6, .(T17, T18)), s(s(T38))) → U4_GA(T6, T17, T18, T38, len120_in_ga(T18, T38))
LEN11_IN_GA(.(T6, .(T17, T18)), s(s(T38))) → LEN120_IN_GA(T18, T38)

The TRS R consists of the following rules:

len11_in_ga([], 0) → len11_out_ga([], 0)
len11_in_ga(.(T6, []), s(0)) → len11_out_ga(.(T6, []), s(0))
len11_in_ga(.(T6, .(T17, T18)), T9) → U3_ga(T6, T17, T18, T9, len120_in_ga(T18, X31))
len120_in_ga([], 0) → len120_out_ga([], 0)
len120_in_ga(.(T24, T25), X49) → U1_ga(T24, T25, X49, len120_in_ga(T25, X48))
len120_in_ga(.(T24, T25), s(T29)) → U2_ga(T24, T25, T29, len120_in_ga(T25, T29))
U2_ga(T24, T25, T29, len120_out_ga(T25, T29)) → len120_out_ga(.(T24, T25), s(T29))
U1_ga(T24, T25, X49, len120_out_ga(T25, X48)) → len120_out_ga(.(T24, T25), X49)
U3_ga(T6, T17, T18, T9, len120_out_ga(T18, X31)) → len11_out_ga(.(T6, .(T17, T18)), T9)
len11_in_ga(.(T6, .(T17, T18)), s(s(T38))) → U4_ga(T6, T17, T18, T38, len120_in_ga(T18, T38))
U4_ga(T6, T17, T18, T38, len120_out_ga(T18, T38)) → len11_out_ga(.(T6, .(T17, T18)), s(s(T38)))

The argument filtering Pi contains the following mapping:
len11_in_ga(x1, x2)  =  len11_in_ga(x1)
[]  =  []
len11_out_ga(x1, x2)  =  len11_out_ga
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
len120_out_ga(x1, x2)  =  len120_out_ga
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
LEN11_IN_GA(x1, x2)  =  LEN11_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(8) Obligation:

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

LEN120_IN_GA(.(T24, T25), s(T29)) → LEN120_IN_GA(T25, T29)
LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)

The TRS R consists of the following rules:

len11_in_ga([], 0) → len11_out_ga([], 0)
len11_in_ga(.(T6, []), s(0)) → len11_out_ga(.(T6, []), s(0))
len11_in_ga(.(T6, .(T17, T18)), T9) → U3_ga(T6, T17, T18, T9, len120_in_ga(T18, X31))
len120_in_ga([], 0) → len120_out_ga([], 0)
len120_in_ga(.(T24, T25), X49) → U1_ga(T24, T25, X49, len120_in_ga(T25, X48))
len120_in_ga(.(T24, T25), s(T29)) → U2_ga(T24, T25, T29, len120_in_ga(T25, T29))
U2_ga(T24, T25, T29, len120_out_ga(T25, T29)) → len120_out_ga(.(T24, T25), s(T29))
U1_ga(T24, T25, X49, len120_out_ga(T25, X48)) → len120_out_ga(.(T24, T25), X49)
U3_ga(T6, T17, T18, T9, len120_out_ga(T18, X31)) → len11_out_ga(.(T6, .(T17, T18)), T9)
len11_in_ga(.(T6, .(T17, T18)), s(s(T38))) → U4_ga(T6, T17, T18, T38, len120_in_ga(T18, T38))
U4_ga(T6, T17, T18, T38, len120_out_ga(T18, T38)) → len11_out_ga(.(T6, .(T17, T18)), s(s(T38)))

The argument filtering Pi contains the following mapping:
len11_in_ga(x1, x2)  =  len11_in_ga(x1)
[]  =  []
len11_out_ga(x1, x2)  =  len11_out_ga
.(x1, x2)  =  .(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
len120_out_ga(x1, x2)  =  len120_out_ga
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

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

LEN120_IN_GA(.(T24, T25), s(T29)) → LEN120_IN_GA(T25, T29)
LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(11) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

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

LEN120_IN_GA(.(T24, T25)) → LEN120_IN_GA(T25)

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

(13) 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:

  • LEN120_IN_GA(.(T24, T25)) → LEN120_IN_GA(T25)
    The graph contains the following edges 1 > 1

(14) YES