(0) Obligation:

Clauses:

len([], X) :- ','(!, eq(X, 0)).
len(Xs, s(N)) :- ','(tail(Xs, Ys), len(Ys, N)).
tail([], []).
tail(.(X, Xs), Xs).
eq(X, X).

Queries:

len(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

len1([], 0).
len1(.(T9, T10), s(T8)) :- len1(T10, T8).

Queries:

len1(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:
len1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

len1_in_ga([], 0) → len1_out_ga([], 0)
len1_in_ga(.(T9, T10), s(T8)) → U1_ga(T9, T10, T8, len1_in_ga(T10, T8))
U1_ga(T9, T10, T8, len1_out_ga(T10, T8)) → len1_out_ga(.(T9, T10), s(T8))

The argument filtering Pi contains the following mapping:
len1_in_ga(x1, x2)  =  len1_in_ga(x1)
[]  =  []
len1_out_ga(x1, x2)  =  len1_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

len1_in_ga([], 0) → len1_out_ga([], 0)
len1_in_ga(.(T9, T10), s(T8)) → U1_ga(T9, T10, T8, len1_in_ga(T10, T8))
U1_ga(T9, T10, T8, len1_out_ga(T10, T8)) → len1_out_ga(.(T9, T10), s(T8))

The argument filtering Pi contains the following mapping:
len1_in_ga(x1, x2)  =  len1_in_ga(x1)
[]  =  []
len1_out_ga(x1, x2)  =  len1_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)

(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:

LEN1_IN_GA(.(T9, T10), s(T8)) → U1_GA(T9, T10, T8, len1_in_ga(T10, T8))
LEN1_IN_GA(.(T9, T10), s(T8)) → LEN1_IN_GA(T10, T8)

The TRS R consists of the following rules:

len1_in_ga([], 0) → len1_out_ga([], 0)
len1_in_ga(.(T9, T10), s(T8)) → U1_ga(T9, T10, T8, len1_in_ga(T10, T8))
U1_ga(T9, T10, T8, len1_out_ga(T10, T8)) → len1_out_ga(.(T9, T10), s(T8))

The argument filtering Pi contains the following mapping:
len1_in_ga(x1, x2)  =  len1_in_ga(x1)
[]  =  []
len1_out_ga(x1, x2)  =  len1_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)
LEN1_IN_GA(x1, x2)  =  LEN1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(6) Obligation:

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

LEN1_IN_GA(.(T9, T10), s(T8)) → U1_GA(T9, T10, T8, len1_in_ga(T10, T8))
LEN1_IN_GA(.(T9, T10), s(T8)) → LEN1_IN_GA(T10, T8)

The TRS R consists of the following rules:

len1_in_ga([], 0) → len1_out_ga([], 0)
len1_in_ga(.(T9, T10), s(T8)) → U1_ga(T9, T10, T8, len1_in_ga(T10, T8))
U1_ga(T9, T10, T8, len1_out_ga(T10, T8)) → len1_out_ga(.(T9, T10), s(T8))

The argument filtering Pi contains the following mapping:
len1_in_ga(x1, x2)  =  len1_in_ga(x1)
[]  =  []
len1_out_ga(x1, x2)  =  len1_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)
LEN1_IN_GA(x1, x2)  =  LEN1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

LEN1_IN_GA(.(T9, T10), s(T8)) → LEN1_IN_GA(T10, T8)

The TRS R consists of the following rules:

len1_in_ga([], 0) → len1_out_ga([], 0)
len1_in_ga(.(T9, T10), s(T8)) → U1_ga(T9, T10, T8, len1_in_ga(T10, T8))
U1_ga(T9, T10, T8, len1_out_ga(T10, T8)) → len1_out_ga(.(T9, T10), s(T8))

The argument filtering Pi contains the following mapping:
len1_in_ga(x1, x2)  =  len1_in_ga(x1)
[]  =  []
len1_out_ga(x1, x2)  =  len1_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)
LEN1_IN_GA(x1, x2)  =  LEN1_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:

LEN1_IN_GA(.(T9, T10), s(T8)) → LEN1_IN_GA(T10, T8)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
LEN1_IN_GA(x1, x2)  =  LEN1_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:

LEN1_IN_GA(.(T9, T10)) → LEN1_IN_GA(T10)

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:

  • LEN1_IN_GA(.(T9, T10)) → LEN1_IN_GA(T10)
    The graph contains the following edges 1 > 1

(14) TRUE