/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

app3_a(g,g,g,a)

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:

app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app([], Ys, Ys).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).


Query: app3_a(g,g,g,a)
----------------------------------------

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

app3_a_in_4: (b,b,b,f)

app_in_3: (b,b,f)

Transforming Prolog into the following Term Rewriting System:

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

   app3_a_in_ggga(Xs, Ys, Zs, Us) -> U1_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U1_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_ggga(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U2_ggga(Xs, Ys, Zs, Us, app_out_gga(Vs, Zs, Us)) -> app3_a_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_a_in_ggga(x1, x2, x3, x4)  =  app3_a_in_ggga(x1, x2, x3)

U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x1, x2, x3, x5)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

[]  =  []

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

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

U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

app3_a_out_ggga(x1, x2, x3, x4)  =  app3_a_out_ggga(x1, x2, x3, x4)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   app3_a_in_ggga(Xs, Ys, Zs, Us) -> U1_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U1_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_ggga(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U2_ggga(Xs, Ys, Zs, Us, app_out_gga(Vs, Zs, Us)) -> app3_a_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_a_in_ggga(x1, x2, x3, x4)  =  app3_a_in_ggga(x1, x2, x3)

U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x1, x2, x3, x5)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

[]  =  []

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

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

U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

app3_a_out_ggga(x1, x2, x3, x4)  =  app3_a_out_ggga(x1, x2, x3, x4)



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

(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:

   APP3_A_IN_GGGA(Xs, Ys, Zs, Us) -> U1_GGGA(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   APP3_A_IN_GGGA(Xs, Ys, Zs, Us) -> APP_IN_GGA(Xs, Ys, Vs)
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U5_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
   U1_GGGA(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_GGGA(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U1_GGGA(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> APP_IN_GGA(Vs, Zs, Us)

The TRS R consists of the following rules:

   app3_a_in_ggga(Xs, Ys, Zs, Us) -> U1_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U1_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_ggga(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U2_ggga(Xs, Ys, Zs, Us, app_out_gga(Vs, Zs, Us)) -> app3_a_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_a_in_ggga(x1, x2, x3, x4)  =  app3_a_in_ggga(x1, x2, x3)

U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x1, x2, x3, x5)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

[]  =  []

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

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

U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

app3_a_out_ggga(x1, x2, x3, x4)  =  app3_a_out_ggga(x1, x2, x3, x4)

APP3_A_IN_GGGA(x1, x2, x3, x4)  =  APP3_A_IN_GGGA(x1, x2, x3)

U1_GGGA(x1, x2, x3, x4, x5)  =  U1_GGGA(x1, x2, x3, x5)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)


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

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

   APP3_A_IN_GGGA(Xs, Ys, Zs, Us) -> U1_GGGA(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   APP3_A_IN_GGGA(Xs, Ys, Zs, Us) -> APP_IN_GGA(Xs, Ys, Vs)
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U5_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
   U1_GGGA(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_GGGA(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U1_GGGA(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> APP_IN_GGA(Vs, Zs, Us)

The TRS R consists of the following rules:

   app3_a_in_ggga(Xs, Ys, Zs, Us) -> U1_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U1_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_ggga(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U2_ggga(Xs, Ys, Zs, Us, app_out_gga(Vs, Zs, Us)) -> app3_a_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_a_in_ggga(x1, x2, x3, x4)  =  app3_a_in_ggga(x1, x2, x3)

U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x1, x2, x3, x5)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

[]  =  []

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

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

U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

app3_a_out_ggga(x1, x2, x3, x4)  =  app3_a_out_ggga(x1, x2, x3, x4)

APP3_A_IN_GGGA(x1, x2, x3, x4)  =  APP3_A_IN_GGGA(x1, x2, x3)

U1_GGGA(x1, x2, x3, x4, x5)  =  U1_GGGA(x1, x2, x3, x5)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)


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

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

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

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

The TRS R consists of the following rules:

   app3_a_in_ggga(Xs, Ys, Zs, Us) -> U1_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Ys, Vs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U1_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Ys, Vs)) -> U2_ggga(Xs, Ys, Zs, Us, app_in_gga(Vs, Zs, Us))
   U2_ggga(Xs, Ys, Zs, Us, app_out_gga(Vs, Zs, Us)) -> app3_a_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_a_in_ggga(x1, x2, x3, x4)  =  app3_a_in_ggga(x1, x2, x3)

U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x1, x2, x3, x5)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

[]  =  []

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

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

U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

app3_a_out_ggga(x1, x2, x3, x4)  =  app3_a_out_ggga(x1, x2, x3, x4)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)


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:

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

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

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)


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:

   APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)

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:
*APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2


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(12)
YES