Left Termination of the query pattern
reverse(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 85 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 22 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 3 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔, 0 ms)
        ↳27 YES

(0) Obligation:

Clauses:

app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
reverse(.(X, Xs), Ys) :- ','(reverse(Xs, Zs), app(Zs, .(X, []), Ys)).
reverse([], []).

Query: reverse(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

reverseA(.(X1, X2), X3) :- reverseA(X2, X4).
reverseA(.(X1, X2), X3) :- ','(reversecA(X2, X4), appB(X4, X1, X3)).
appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
appC(.(X1, X2), X3, .(X1, X4)) :- appC(X2, X3, X4).
reverseD(.(X1, .(X2, X3)), X4) :- reverseA(X3, X5).
reverseD(.(X1, .(X2, X3)), X4) :- ','(reversecA(X3, X5), appB(X5, X2, X6)).
reverseD(.(X1, .(X2, X3)), X4) :- ','(reversecA(X3, X5), ','(appcB(X5, X2, X6), appC(X6, X1, X4))).

Clauses:

reversecA(.(X1, X2), X3) :- ','(reversecA(X2, X4), appcB(X4, X1, X3)).
reversecA([], []).
appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
appcB([], X1, .(X1, [])).
appcC(.(X1, X2), X3, .(X1, X4)) :- appcC(X2, X3, X4).
appcC([], X1, .(X1, [])).

Afs:

reverseD(x1, x2)  =  reverseD(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
reverseD_in: (b,f)
reverseA_in: (b,f)
reversecA_in: (b,f)
appcB_in: (b,b,f)
appB_in: (b,b,f)
appC_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVERSED_IN_GA(.(X1, .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, reverseA_in_ga(X3, X5))
REVERSED_IN_GA(.(X1, .(X2, X3)), X4) → REVERSEA_IN_GA(X3, X5)
REVERSEA_IN_GA(.(X1, X2), X3) → U1_GA(X1, X2, X3, reverseA_in_ga(X2, X4))
REVERSEA_IN_GA(.(X1, X2), X3) → REVERSEA_IN_GA(X2, X4)
REVERSEA_IN_GA(.(X1, X2), X3) → U2_GA(X1, X2, X3, reversecA_in_ga(X2, X4))
U2_GA(X1, X2, X3, reversecA_out_ga(X2, X4)) → U3_GA(X1, X2, X3, appB_in_gga(X4, X1, X3))
U2_GA(X1, X2, X3, reversecA_out_ga(X2, X4)) → APPB_IN_GGA(X4, X1, X3)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U4_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
REVERSED_IN_GA(.(X1, .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, reversecA_in_ga(X3, X5))
U7_GA(X1, X2, X3, X4, reversecA_out_ga(X3, X5)) → U8_GA(X1, X2, X3, X4, appB_in_gga(X5, X2, X6))
U7_GA(X1, X2, X3, X4, reversecA_out_ga(X3, X5)) → APPB_IN_GGA(X5, X2, X6)
U7_GA(X1, X2, X3, X4, reversecA_out_ga(X3, X5)) → U9_GA(X1, X2, X3, X4, appcB_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → U10_GA(X1, X2, X3, X4, appC_in_gga(X6, X1, X4))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → APPC_IN_GGA(X6, X1, X4)
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U5_GGA(X1, X2, X3, X4, appC_in_gga(X2, X3, X4))
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

reversecA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, reversecA_in_ga(X2, X4))
reversecA_in_ga([], []) → reversecA_out_ga([], [])
U12_ga(X1, X2, X3, reversecA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → reversecA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reverseA_in_ga(x1, x2)  =  reverseA_in_ga(x1)
reversecA_in_ga(x1, x2)  =  reversecA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversecA_out_ga(x1, x2)  =  reversecA_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appC_in_gga(x1, x2, x3)  =  appC_in_gga(x1, x2)
REVERSED_IN_GA(x1, x2)  =  REVERSED_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REVERSEA_IN_GA(x1, x2)  =  REVERSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
APPC_IN_GGA(x1, x2, x3)  =  APPC_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

REVERSED_IN_GA(.(X1, .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, reverseA_in_ga(X3, X5))
REVERSED_IN_GA(.(X1, .(X2, X3)), X4) → REVERSEA_IN_GA(X3, X5)
REVERSEA_IN_GA(.(X1, X2), X3) → U1_GA(X1, X2, X3, reverseA_in_ga(X2, X4))
REVERSEA_IN_GA(.(X1, X2), X3) → REVERSEA_IN_GA(X2, X4)
REVERSEA_IN_GA(.(X1, X2), X3) → U2_GA(X1, X2, X3, reversecA_in_ga(X2, X4))
U2_GA(X1, X2, X3, reversecA_out_ga(X2, X4)) → U3_GA(X1, X2, X3, appB_in_gga(X4, X1, X3))
U2_GA(X1, X2, X3, reversecA_out_ga(X2, X4)) → APPB_IN_GGA(X4, X1, X3)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U4_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
REVERSED_IN_GA(.(X1, .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, reversecA_in_ga(X3, X5))
U7_GA(X1, X2, X3, X4, reversecA_out_ga(X3, X5)) → U8_GA(X1, X2, X3, X4, appB_in_gga(X5, X2, X6))
U7_GA(X1, X2, X3, X4, reversecA_out_ga(X3, X5)) → APPB_IN_GGA(X5, X2, X6)
U7_GA(X1, X2, X3, X4, reversecA_out_ga(X3, X5)) → U9_GA(X1, X2, X3, X4, appcB_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → U10_GA(X1, X2, X3, X4, appC_in_gga(X6, X1, X4))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → APPC_IN_GGA(X6, X1, X4)
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U5_GGA(X1, X2, X3, X4, appC_in_gga(X2, X3, X4))
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

reversecA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, reversecA_in_ga(X2, X4))
reversecA_in_ga([], []) → reversecA_out_ga([], [])
U12_ga(X1, X2, X3, reversecA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → reversecA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reverseA_in_ga(x1, x2)  =  reverseA_in_ga(x1)
reversecA_in_ga(x1, x2)  =  reversecA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversecA_out_ga(x1, x2)  =  reversecA_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appC_in_gga(x1, x2, x3)  =  appC_in_gga(x1, x2)
REVERSED_IN_GA(x1, x2)  =  REVERSED_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REVERSEA_IN_GA(x1, x2)  =  REVERSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
APPC_IN_GGA(x1, x2, x3)  =  APPC_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

reversecA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, reversecA_in_ga(X2, X4))
reversecA_in_ga([], []) → reversecA_out_ga([], [])
U12_ga(X1, X2, X3, reversecA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → reversecA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reversecA_in_ga(x1, x2)  =  reversecA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversecA_out_ga(x1, x2)  =  reversecA_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
APPC_IN_GGA(x1, x2, x3)  =  APPC_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPC_IN_GGA(x1, x2, x3)  =  APPC_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APPC_IN_GGA(.(X1, X2), X3) → APPC_IN_GGA(X2, X3)

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

(12) 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:

  • APPC_IN_GGA(.(X1, X2), X3) → APPC_IN_GGA(X2, X3)
    The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

reversecA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, reversecA_in_ga(X2, X4))
reversecA_in_ga([], []) → reversecA_out_ga([], [])
U12_ga(X1, X2, X3, reversecA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → reversecA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reversecA_in_ga(x1, x2)  =  reversecA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversecA_out_ga(x1, x2)  =  reversecA_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)

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

(19) 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:

  • APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)
    The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

REVERSEA_IN_GA(.(X1, X2), X3) → REVERSEA_IN_GA(X2, X4)

The TRS R consists of the following rules:

reversecA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, reversecA_in_ga(X2, X4))
reversecA_in_ga([], []) → reversecA_out_ga([], [])
U12_ga(X1, X2, X3, reversecA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → reversecA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reversecA_in_ga(x1, x2)  =  reversecA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversecA_out_ga(x1, x2)  =  reversecA_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
REVERSEA_IN_GA(x1, x2)  =  REVERSEA_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

REVERSEA_IN_GA(.(X1, X2), X3) → REVERSEA_IN_GA(X2, X4)

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

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

REVERSEA_IN_GA(.(X1, X2)) → REVERSEA_IN_GA(X2)

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

(26) 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:

  • REVERSEA_IN_GA(.(X1, X2)) → REVERSEA_IN_GA(X2)
    The graph contains the following edges 1 > 1

(27) YES