(0) Obligation:
Clauses:p(val_i, val_j).
map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys)).
map([], []).
Queries:
map(g,a).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:map1(.(val_i, T8), .(val_j, T13)) :- map1(T8, T13).
map1([], []).
Queries:
map1(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:map1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
map1_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, map1_in_ga(T8, T13))
map1_in_ga([], []) → map1_out_ga([], [])
U1_ga(T8, T13, map1_out_ga(T8, T13)) → map1_out_ga(.(val_i, T8), .(val_j, T13))
The argument filtering Pi contains the following mapping:
map1_in_ga(x1, x2) = map1_in_ga(x1)
.(x1, x2) = .(x1, x2)
val_i = val_i
U1_ga(x1, x2, x3) = U1_ga(x3)
[] = []
map1_out_ga(x1, x2) = map1_out_ga(x2)
val_j = val_j
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
map1_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, map1_in_ga(T8, T13))
map1_in_ga([], []) → map1_out_ga([], [])
U1_ga(T8, T13, map1_out_ga(T8, T13)) → map1_out_ga(.(val_i, T8), .(val_j, T13))
The argument filtering Pi contains the following mapping:
map1_in_ga(x1, x2) = map1_in_ga(x1)
.(x1, x2) = .(x1, x2)
val_i = val_i
U1_ga(x1, x2, x3) = U1_ga(x3)
[] = []
map1_out_ga(x1, x2) = map1_out_ga(x2)
val_j = val_j
(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:
MAP1_IN_GA(.(val_i, T8), .(val_j, T13)) → U1_GA(T8, T13, map1_in_ga(T8, T13))
MAP1_IN_GA(.(val_i, T8), .(val_j, T13)) → MAP1_IN_GA(T8, T13)
The TRS R consists of the following rules:
map1_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, map1_in_ga(T8, T13))
map1_in_ga([], []) → map1_out_ga([], [])
U1_ga(T8, T13, map1_out_ga(T8, T13)) → map1_out_ga(.(val_i, T8), .(val_j, T13))
The argument filtering Pi contains the following mapping:
map1_in_ga(x1, x2) = map1_in_ga(x1)
.(x1, x2) = .(x1, x2)
val_i = val_i
U1_ga(x1, x2, x3) = U1_ga(x3)
[] = []
map1_out_ga(x1, x2) = map1_out_ga(x2)
val_j = val_j
MAP1_IN_GA(x1, x2) = MAP1_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MAP1_IN_GA(.(val_i, T8), .(val_j, T13)) → U1_GA(T8, T13, map1_in_ga(T8, T13))
MAP1_IN_GA(.(val_i, T8), .(val_j, T13)) → MAP1_IN_GA(T8, T13)
The TRS R consists of the following rules:
map1_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, map1_in_ga(T8, T13))
map1_in_ga([], []) → map1_out_ga([], [])
U1_ga(T8, T13, map1_out_ga(T8, T13)) → map1_out_ga(.(val_i, T8), .(val_j, T13))
The argument filtering Pi contains the following mapping:
map1_in_ga(x1, x2) = map1_in_ga(x1)
.(x1, x2) = .(x1, x2)
val_i = val_i
U1_ga(x1, x2, x3) = U1_ga(x3)
[] = []
map1_out_ga(x1, x2) = map1_out_ga(x2)
val_j = val_j
MAP1_IN_GA(x1, x2) = MAP1_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
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:
MAP1_IN_GA(.(val_i, T8), .(val_j, T13)) → MAP1_IN_GA(T8, T13)
The TRS R consists of the following rules:
map1_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, map1_in_ga(T8, T13))
map1_in_ga([], []) → map1_out_ga([], [])
U1_ga(T8, T13, map1_out_ga(T8, T13)) → map1_out_ga(.(val_i, T8), .(val_j, T13))
The argument filtering Pi contains the following mapping:
map1_in_ga(x1, x2) = map1_in_ga(x1)
.(x1, x2) = .(x1, x2)
val_i = val_i
U1_ga(x1, x2, x3) = U1_ga(x3)
[] = []
map1_out_ga(x1, x2) = map1_out_ga(x2)
val_j = val_j
MAP1_IN_GA(x1, x2) = MAP1_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:
MAP1_IN_GA(.(val_i, T8), .(val_j, T13)) → MAP1_IN_GA(T8, T13)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
val_i = val_i
val_j = val_j
MAP1_IN_GA(x1, x2) = MAP1_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:
MAP1_IN_GA(.(val_i, T8)) → MAP1_IN_GA(T8)
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:
- MAP1_IN_GA(.(val_i, T8)) → MAP1_IN_GA(T8)
The graph contains the following edges 1 > 1