Left Termination of the query pattern append3_in_3(a, 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:

append1([], X, X).
append1(.(X, Y), U, .(X, Z)) :- append1(Y, U, Z).
append2([], X, X).
append2(.(X, Y), U, .(X, Z)) :- append2(Y, U, Z).
append3([], X, X).
append3(.(X, Y), U, .(X, Z)) :- append3(Y, U, Z).

Queries:

append3(a,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:
append3_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

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_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

(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:

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → U3_AAG(X, Y, U, Z, append3_in_aag(Y, U, Z))
APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_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:

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → U3_AAG(X, Y, U, Z, append3_in_aag(Y, U, Z))
APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_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 1 less node.

(6) Obligation:

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

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_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:

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_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:

APPEND3_IN_AAG(.(X, Z)) → APPEND3_IN_AAG(Z)

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:

  • APPEND3_IN_AAG(.(X, Z)) → APPEND3_IN_AAG(Z)
    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:
append3_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)

(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:

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → U3_AAG(X, Y, U, Z, append3_in_aag(Y, U, Z))
APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_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:

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → U3_AAG(X, Y, U, Z, append3_in_aag(Y, U, Z))
APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_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 1 less node.

(18) Obligation:

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

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

The TRS R consists of the following rules:

append3_in_aag([], X, X) → append3_out_aag([], X, X)
append3_in_aag(.(X, Y), U, .(X, Z)) → U3_aag(X, Y, U, Z, append3_in_aag(Y, U, Z))
U3_aag(X, Y, U, Z, append3_out_aag(Y, U, Z)) → append3_out_aag(.(X, Y), U, .(X, Z))

The argument filtering Pi contains the following mapping:
append3_in_aag(x1, x2, x3)  =  append3_in_aag(x3)
append3_out_aag(x1, x2, x3)  =  append3_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_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:

APPEND3_IN_AAG(.(X, Y), U, .(X, Z)) → APPEND3_IN_AAG(Y, U, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND3_IN_AAG(x1, x2, x3)  =  APPEND3_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:

APPEND3_IN_AAG(.(X, Z)) → APPEND3_IN_AAG(Z)

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