/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

p(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


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(0)
Obligation:
Clauses:

p(M, N, s(R), RES) :- p(M, R, N, RES).
p(M, s(N), R, RES) :- p(R, N, M, RES).
p(M, X1, X2, M).


Query: p(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:

p_in_4: (b,b,b,f)

Transforming Prolog into the following Term Rewriting System:

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

   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)

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

s(x1)  =  s(x1)

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

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

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





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)

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

s(x1)  =  s(x1)

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

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

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



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(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:

   P_IN_GGGA(M, N, s(R), RES) -> U1_GGGA(M, N, R, RES, p_in_ggga(M, R, N, RES))
   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)
   P_IN_GGGA(M, s(N), R, RES) -> U2_GGGA(M, N, R, RES, p_in_ggga(R, N, M, RES))
   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)

The TRS R consists of the following rules:

   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)

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

s(x1)  =  s(x1)

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

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

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

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

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

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


We have to consider all (P,R,Pi)-chains
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(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   P_IN_GGGA(M, N, s(R), RES) -> U1_GGGA(M, N, R, RES, p_in_ggga(M, R, N, RES))
   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)
   P_IN_GGGA(M, s(N), R, RES) -> U2_GGGA(M, N, R, RES, p_in_ggga(R, N, M, RES))
   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)

The TRS R consists of the following rules:

   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)

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

s(x1)  =  s(x1)

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

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

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

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

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

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


We have to consider all (P,R,Pi)-chains
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
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(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)
   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)

The TRS R consists of the following rules:

   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)

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

s(x1)  =  s(x1)

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

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

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

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


We have to consider all (P,R,Pi)-chains
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(7) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
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(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)
   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)

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

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


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

   P_IN_GGGA(M, s(N), R) -> P_IN_GGGA(R, N, M)
   P_IN_GGGA(M, N, s(R)) -> P_IN_GGGA(M, R, N)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
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(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:
*P_IN_GGGA(M, s(N), R) -> P_IN_GGGA(R, N, M)
The graph contains the following edges 3 >= 1, 2 > 2, 1 >= 3


*P_IN_GGGA(M, N, s(R)) -> P_IN_GGGA(M, R, N)
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3


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(12)
YES