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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

append3(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:

append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).


Query: append3(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:

append3_in_4: (b,b,b,f)

append_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:

   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
   append_in_gga([], L, L) -> append_out_gga([], L, L)
   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)

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

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)

append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(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:

   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
   append_in_gga([], L, L) -> append_out_gga([], L, L)
   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)

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

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)

append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(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:

   APPEND3_IN_GGGA(A, B, C, D) -> U2_GGGA(A, B, C, D, append_in_gga(A, B, E))
   APPEND3_IN_GGGA(A, B, C, D) -> APPEND_IN_GGA(A, B, E)
   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U1_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> U3_GGGA(A, B, C, D, append_in_gga(E, C, D))
   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> APPEND_IN_GGA(E, C, D)

The TRS R consists of the following rules:

   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
   append_in_gga([], L, L) -> append_out_gga([], L, L)
   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)

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

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)

append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)

APPEND3_IN_GGGA(x1, x2, x3, x4)  =  APPEND3_IN_GGGA(x1, x2, x3)

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

APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)

U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)

U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)


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

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

   APPEND3_IN_GGGA(A, B, C, D) -> U2_GGGA(A, B, C, D, append_in_gga(A, B, E))
   APPEND3_IN_GGGA(A, B, C, D) -> APPEND_IN_GGA(A, B, E)
   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U1_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> U3_GGGA(A, B, C, D, append_in_gga(E, C, D))
   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> APPEND_IN_GGA(E, C, D)

The TRS R consists of the following rules:

   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
   append_in_gga([], L, L) -> append_out_gga([], L, L)
   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)

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

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)

append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)

APPEND3_IN_GGGA(x1, x2, x3, x4)  =  APPEND3_IN_GGGA(x1, x2, x3)

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

APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)

U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)

U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(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:

   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)

The TRS R consists of the following rules:

   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
   append_in_gga([], L, L) -> append_out_gga([], L, L)
   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)

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

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)

append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)

APPEND_IN_GGA(x1, x2, x3)  =  APPEND_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:

   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)

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

APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)


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

   APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2)

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:
*APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2)
The graph contains the following edges 1 > 1, 2 >= 2


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(12)
YES