Left Termination of the query pattern prefix_in_2(a, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 QDPSizeChangeProof (⇔)
    ↳12 TRUE
  ↳13 PrologToPiTRSProof (⇐)
    ↳14 PiTRS
    ↳15 DependencyPairsProof (⇔)
    ↳16 PiDP
    ↳17 DependencyGraphProof (⇔)
    ↳18 PiDP
    ↳19 UsableRulesProof (⇔)
    ↳20 PiDP
    ↳21 PiDPToQDPProof (⇐)
    ↳22 QDP

(0) Obligation:

Clauses:

prefix(Xs, Ys) :- app(Xs, X1, Ys).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

prefix(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
prefix_in: (f,b)
app_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

PREFIX_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, app_in_aag(Xs, X1, Ys))
PREFIX_IN_AG(Xs, Ys) → APP_IN_AAG(Xs, X1, Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1)
PREFIX_IN_AG(x1, x2)  =  PREFIX_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x5)

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

(4) Obligation:

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

PREFIX_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, app_in_aag(Xs, X1, Ys))
PREFIX_IN_AG(Xs, Ys) → APP_IN_AAG(Xs, X1, Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1)
PREFIX_IN_AG(x1, x2)  =  PREFIX_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

(11) 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:

  • APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
    The graph contains the following edges 1 > 1

(12) TRUE

(13) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
prefix_in: (f,b)
app_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1, x2)

(15) DependencyPairsProof (EQUIVALENT transformation)

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

PREFIX_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, app_in_aag(Xs, X1, Ys))
PREFIX_IN_AG(Xs, Ys) → APP_IN_AAG(Xs, X1, Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1, x2)
PREFIX_IN_AG(x1, x2)  =  PREFIX_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)

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

(16) Obligation:

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

PREFIX_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, app_in_aag(Xs, X1, Ys))
PREFIX_IN_AG(Xs, Ys) → APP_IN_AAG(Xs, X1, Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1, x2)
PREFIX_IN_AG(x1, x2)  =  PREFIX_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(18) Obligation:

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

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

prefix_in_ag(Xs, Ys) → U1_ag(Xs, Ys, app_in_aag(Xs, X1, Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U2_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ag(Xs, Ys, app_out_aag(Xs, X1, Ys)) → prefix_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
prefix_in_ag(x1, x2)  =  prefix_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
prefix_out_ag(x1, x2)  =  prefix_out_ag(x1, x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

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

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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