(0) Obligation:
Clauses:f([], RES, RES).
f(.(Head, Tail), X, RES) :- g(Tail, X, .(Head, Tail), RES).
g(A, B, C, RES) :- f(A, .(B, C), RES).
Queries:
f(g,g,a).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:f1([], T5, T5).
f1(.(T30, T28), T29, T32) :- f1(T28, .(T29, .(T30, T28)), T32).
Queries:
f1(g,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:f1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
f1_in_gga([], T5, T5) → f1_out_gga([], T5, T5)
f1_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, f1_out_gga(T28, .(T29, .(T30, T28)), T32)) → f1_out_gga(.(T30, T28), T29, T32)
The argument filtering Pi contains the following mapping:
f1_in_gga(x1, x2, x3) = f1_in_gga(x1, x2)
[] = []
f1_out_gga(x1, x2, x3) = f1_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(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:
f1_in_gga([], T5, T5) → f1_out_gga([], T5, T5)
f1_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, f1_out_gga(T28, .(T29, .(T30, T28)), T32)) → f1_out_gga(.(T30, T28), T29, T32)
The argument filtering Pi contains the following mapping:
f1_in_gga(x1, x2, x3) = f1_in_gga(x1, x2)
[] = []
f1_out_gga(x1, x2, x3) = f1_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(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:
F1_IN_GGA(.(T30, T28), T29, T32) → U1_GGA(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
F1_IN_GGA(.(T30, T28), T29, T32) → F1_IN_GGA(T28, .(T29, .(T30, T28)), T32)
The TRS R consists of the following rules:
f1_in_gga([], T5, T5) → f1_out_gga([], T5, T5)
f1_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, f1_out_gga(T28, .(T29, .(T30, T28)), T32)) → f1_out_gga(.(T30, T28), T29, T32)
The argument filtering Pi contains the following mapping:
f1_in_gga(x1, x2, x3) = f1_in_gga(x1, x2)
[] = []
f1_out_gga(x1, x2, x3) = f1_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
F1_IN_GGA(x1, x2, x3) = F1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
F1_IN_GGA(.(T30, T28), T29, T32) → U1_GGA(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
F1_IN_GGA(.(T30, T28), T29, T32) → F1_IN_GGA(T28, .(T29, .(T30, T28)), T32)
The TRS R consists of the following rules:
f1_in_gga([], T5, T5) → f1_out_gga([], T5, T5)
f1_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, f1_out_gga(T28, .(T29, .(T30, T28)), T32)) → f1_out_gga(.(T30, T28), T29, T32)
The argument filtering Pi contains the following mapping:
f1_in_gga(x1, x2, x3) = f1_in_gga(x1, x2)
[] = []
f1_out_gga(x1, x2, x3) = f1_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
F1_IN_GGA(x1, x2, x3) = F1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(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 1 less node.(8) Obligation:
Pi DP problem:The TRS P consists of the following rules:
F1_IN_GGA(.(T30, T28), T29, T32) → F1_IN_GGA(T28, .(T29, .(T30, T28)), T32)
The TRS R consists of the following rules:
f1_in_gga([], T5, T5) → f1_out_gga([], T5, T5)
f1_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, f1_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, f1_out_gga(T28, .(T29, .(T30, T28)), T32)) → f1_out_gga(.(T30, T28), T29, T32)
The argument filtering Pi contains the following mapping:
f1_in_gga(x1, x2, x3) = f1_in_gga(x1, x2)
[] = []
f1_out_gga(x1, x2, x3) = f1_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
F1_IN_GGA(x1, x2, x3) = F1_IN_GGA(x1, x2)
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:
F1_IN_GGA(.(T30, T28), T29, T32) → F1_IN_GGA(T28, .(T29, .(T30, T28)), T32)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
F1_IN_GGA(x1, x2, x3) = F1_IN_GGA(x1, x2)
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:
F1_IN_GGA(.(T30, T28), T29) → F1_IN_GGA(T28, .(T29, .(T30, T28)))
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:
- F1_IN_GGA(.(T30, T28), T29) → F1_IN_GGA(T28, .(T29, .(T30, T28)))
The graph contains the following edges 1 > 1