/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Left Termination of the query pattern

app2(a,g,g)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) PiDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) PiDP
(9) PiDPToQDPProof [SOUND, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


----------------------------------------

(0)
Obligation:
Clauses:

app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).


Query: app2(a,g,g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

app2_in_3: (f,b,b)

Transforming Prolog into the following Term Rewriting System:

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

   app2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   app2_in_agg([], Ys, Ys) -> app2_out_agg([], Ys, Ys)
   U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) -> app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)

app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   app2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   app2_in_agg([], Ys, Ys) -> app2_out_agg([], Ys, Ys)
   U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) -> app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)

app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)



----------------------------------------

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

   APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> U2_AGG(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   app2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   app2_in_agg([], Ys, Ys) -> app2_out_agg([], Ys, Ys)
   U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) -> app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)

app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)

APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)

U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x3, x4, x5)


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

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> U2_AGG(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   app2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   app2_in_agg([], Ys, Ys) -> app2_out_agg([], Ys, Ys)
   U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) -> app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)

app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)

APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)

U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x3, x4, x5)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
----------------------------------------

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   app2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
   app2_in_agg([], Ys, Ys) -> app2_out_agg([], Ys, Ys)
   U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) -> app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)

app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)

APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)


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

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

(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AGG(Xs, Ys, Zs)

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

APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)


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

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP2_IN_AGG(Ys, .(X, Zs)) -> APP2_IN_AGG(Ys, Zs)

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

(11) QDPSizeChangeProof (EQUIVALENT)
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:
*APP2_IN_AGG(Ys, .(X, Zs)) -> APP2_IN_AGG(Ys, Zs)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(12)
YES