Left Termination of the query pattern fold_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

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

(0) Obligation:

Clauses:

fold(X, [], Z) :- ','(!, eq(X, Z)).
fold(X, Y, Z) :- ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z)))).
myop(a, b, c).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).

Queries:

fold(g,g,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

fold(X, [], Z) :- eq(X, Z).
fold(X, Y, Z) :- ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z)))).
myop(a, b, c).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).

Queries:

fold(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

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

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x1, x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x1, x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)

(5) 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:

FOLD_IN_GGA(X, [], Z) → U1_GGA(X, Z, eq_in_ga(X, Z))
FOLD_IN_GGA(X, [], Z) → EQ_IN_GA(X, Z)
FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
FOLD_IN_GGA(X, Y, Z) → HEAD_IN_GA(Y, H)
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → MYOP_IN_GAA(X, H, V)
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_GGA(X, Y, Z, fold_in_gga(V, T, Z))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x1, x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x5, x6)
MYOP_IN_GAA(x1, x2, x3)  =  MYOP_IN_GAA(x1)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

(6) Obligation:

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

FOLD_IN_GGA(X, [], Z) → U1_GGA(X, Z, eq_in_ga(X, Z))
FOLD_IN_GGA(X, [], Z) → EQ_IN_GA(X, Z)
FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
FOLD_IN_GGA(X, Y, Z) → HEAD_IN_GA(Y, H)
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → MYOP_IN_GAA(X, H, V)
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_GGA(X, Y, Z, fold_in_gga(V, T, Z))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x1, x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x5, x6)
MYOP_IN_GAA(x1, x2, x3)  =  MYOP_IN_GAA(x1)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(8) Obligation:

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

FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x1, x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x5)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x5, x6)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)

The argument filtering Pi contains the following mapping:
[]  =  []
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x1, x2, x3)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x5)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x5, x6)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

FOLD_IN_GGA(X, Y) → U2_GGA(X, Y, head_in_ga(Y))
U2_GGA(X, Y, head_out_ga(Y)) → U3_GGA(X, Y, tail_in_ga(Y))
U3_GGA(X, Y, tail_out_ga(Y, T)) → U4_GGA(X, Y, T, myop_in_gaa(X))
U4_GGA(X, Y, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(H, X2)) → head_out_ga(.(H, X2))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X3, T)) → tail_out_ga(.(X3, T), T)
myop_in_gaa(a) → myop_out_gaa(a, b, c)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)
myop_in_gaa(x0)

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

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

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

(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:

FOLD_IN_GGA(X, [], Z) → U1_GGA(X, Z, eq_in_ga(X, Z))
FOLD_IN_GGA(X, [], Z) → EQ_IN_GA(X, Z)
FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
FOLD_IN_GGA(X, Y, Z) → HEAD_IN_GA(Y, H)
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → MYOP_IN_GAA(X, H, V)
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_GGA(X, Y, Z, fold_in_gga(V, T, Z))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x5, x6)
MYOP_IN_GAA(x1, x2, x3)  =  MYOP_IN_GAA(x1)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(16) Obligation:

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

FOLD_IN_GGA(X, [], Z) → U1_GGA(X, Z, eq_in_ga(X, Z))
FOLD_IN_GGA(X, [], Z) → EQ_IN_GA(X, Z)
FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
FOLD_IN_GGA(X, Y, Z) → HEAD_IN_GA(Y, H)
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T)
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → MYOP_IN_GAA(X, H, V)
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_GGA(X, Y, Z, fold_in_gga(V, T, Z))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x5, x6)
MYOP_IN_GAA(x1, x2, x3)  =  MYOP_IN_GAA(x1)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(18) Obligation:

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

FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

fold_in_gga(X, [], Z) → U1_gga(X, Z, eq_in_ga(X, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U1_gga(X, Z, eq_out_ga(X, Z)) → fold_out_gga(X, [], Z)
fold_in_gga(X, Y, Z) → U2_gga(X, Y, Z, head_in_ga(Y, H))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
U2_gga(X, Y, Z, head_out_ga(Y, H)) → U3_gga(X, Y, Z, H, tail_in_ga(Y, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
U3_gga(X, Y, Z, H, tail_out_ga(Y, T)) → U4_gga(X, Y, Z, H, T, myop_in_gaa(X, H, V))
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)
U4_gga(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → U5_gga(X, Y, Z, fold_in_gga(V, T, Z))
U5_gga(X, Y, Z, fold_out_gga(V, T, Z)) → fold_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x5, x6)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x2, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x5, x6)

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:

FOLD_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, head_in_ga(Y, H))
U2_GGA(X, Y, Z, head_out_ga(Y, H)) → U3_GGA(X, Y, Z, H, tail_in_ga(Y, T))
U3_GGA(X, Y, Z, H, tail_out_ga(Y, T)) → U4_GGA(X, Y, Z, H, T, myop_in_gaa(X, H, V))
U4_GGA(X, Y, Z, H, T, myop_out_gaa(X, H, V)) → FOLD_IN_GGA(V, T, Z)

The TRS R consists of the following rules:

head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(H, X2), H) → head_out_ga(.(H, X2), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, T), T) → tail_out_ga(.(X3, T), T)
myop_in_gaa(a, b, c) → myop_out_gaa(a, b, c)

The argument filtering Pi contains the following mapping:
[]  =  []
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
myop_in_gaa(x1, x2, x3)  =  myop_in_gaa(x1)
a  =  a
myop_out_gaa(x1, x2, x3)  =  myop_out_gaa(x2, x3)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x5, x6)

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:

FOLD_IN_GGA(X, Y) → U2_GGA(X, Y, head_in_ga(Y))
U2_GGA(X, Y, head_out_ga) → U3_GGA(X, tail_in_ga(Y))
U3_GGA(X, tail_out_ga(T)) → U4_GGA(T, myop_in_gaa(X))
U4_GGA(T, myop_out_gaa(H, V)) → FOLD_IN_GGA(V, T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(H, X2)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(X3, T)) → tail_out_ga(T)
myop_in_gaa(a) → myop_out_gaa(b, c)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)
myop_in_gaa(x0)

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

(23) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

myop_in_gaa(a) → myop_out_gaa(b, c)
tail_in_ga(.(X3, T)) → tail_out_ga(T)
head_in_ga(.(H, X2)) → head_out_ga
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + 2·x2   
POL(FOLD_IN_GGA(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U2_GGA(x1, x2, x3)) = 1 + 2·x1 + x2 + x3   
POL(U3_GGA(x1, x2)) = x1 + x2   
POL(U4_GGA(x1, x2)) = 2·x1 + x2   
POL([]) = 1   
POL(a) = 1   
POL(b) = 0   
POL(c) = 0   
POL(head_in_ga(x1)) = x1   
POL(head_out_ga) = 1   
POL(myop_in_gaa(x1)) = 1 + x1   
POL(myop_out_gaa(x1, x2)) = 1 + x1 + 2·x2   
POL(tail_in_ga(x1)) = 2 + x1   
POL(tail_out_ga(x1)) = 1 + 2·x1   

(24) Obligation:

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

FOLD_IN_GGA(X, Y) → U2_GGA(X, Y, head_in_ga(Y))
U2_GGA(X, Y, head_out_ga) → U3_GGA(X, tail_in_ga(Y))
U3_GGA(X, tail_out_ga(T)) → U4_GGA(T, myop_in_gaa(X))
U4_GGA(T, myop_out_gaa(H, V)) → FOLD_IN_GGA(V, T)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
head_in_ga([]) → head_out_ga

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)
myop_in_gaa(x0)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(26) TRUE