Left Termination proof of /tmp/tmpLro8Lr/star1.pl

Left Termination of the query pattern star_in_2(g, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 CutEliminatorProof (⇐)

↳2 Prolog

↳3 UndefinedPredicateHandlerProof (⇐)

↳4 Prolog

  ↳5 PrologToPiTRSProof (⇐)

    ↳6 PiTRS

    ↳7 DependencyPairsProof (⇔)

    ↳8 PiDP

    ↳9 DependencyGraphProof (⇔)

    ↳10 AND

      ↳11 PiDP

        ↳12 UsableRulesProof (⇔)

        ↳13 PiDP

        ↳14 PiDPToQDPProof (⇐)

        ↳15 QDP

        ↳16 QDPSizeChangeProof (⇔)

        ↳17 TRUE

      ↳18 PiDP

        ↳19 UsableRulesProof (⇔)

        ↳20 PiDP

        ↳21 PiDPToQDPProof (⇐)

        ↳22 QDP

        ↳23 Narrowing (⇐)

        ↳24 QDP

        ↳25 Instantiation (⇔)

        ↳26 QDP

        ↳27 DependencyGraphProof (⇔)

        ↳28 AND

          ↳29 QDP

            ↳30 ForwardInstantiation (⇔)

            ↳31 QDP

            ↳32 QDPOrderProof (⇔)

            ↳33 QDP

            ↳34 DependencyGraphProof (⇔)

            ↳35 TRUE

          ↳36 QDP

            ↳37 UsableRulesProof (⇔)

            ↳38 QDP

            ↳39 QReductionProof (⇔)

            ↳40 QDP

            ↳41 NonTerminationProof (⇔)

            ↳42 FALSE

  ↳43 PrologToPiTRSProof (⇐)

    ↳44 PiTRS

    ↳45 DependencyPairsProof (⇔)

    ↳46 PiDP

    ↳47 DependencyGraphProof (⇔)

    ↳48 AND

      ↳49 PiDP

        ↳50 UsableRulesProof (⇔)

        ↳51 PiDP

        ↳52 PiDPToQDPProof (⇐)

        ↳53 QDP

        ↳54 QDPSizeChangeProof (⇔)

        ↳55 TRUE

      ↳56 PiDP

        ↳57 UsableRulesProof (⇔)

        ↳58 PiDP

        ↳59 PiDPToQDPProof (⇐)

        ↳60 QDP

        ↳61 Narrowing (⇐)

        ↳62 QDP

        ↳63 Instantiation (⇔)

        ↳64 QDP

        ↳65 DependencyGraphProof (⇔)

        ↳66 AND

          ↳67 QDP

            ↳68 ForwardInstantiation (⇔)

            ↳69 QDP

            ↳70 QDPOrderProof (⇔)

            ↳71 QDP

            ↳72 DependencyGraphProof (⇔)

            ↳73 TRUE

          ↳74 QDP

            ↳75 UsableRulesProof (⇔)

            ↳76 QDP

            ↳77 QReductionProof (⇔)

            ↳78 QDP

            ↳79 NonTerminationProof (⇔)

            ↳80 FALSE

(0) Obligation:

Clauses:

star(U, []) :- !.
star([], W) :- ','(!, =(W, [])).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
=(X, X).

Queries:

star(g,g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

star(U, []) :- true.
star([], W) :- =(W, []).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
=(X, X).

Queries:

star(g,g).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

star(U, []) :- true.
star([], W) :- =(W, []).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
=(X, X).
true.

Queries:

star(g,g).

(5) PrologToPiTRSProof (SOUND transformation)

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

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

(7) 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:

STAR_IN_GG(U, []) → U1_GG(U, true_in_)
STAR_IN_GG(U, []) → TRUE_IN_
STAR_IN_GG([], W) → U2_GG(W, =_in_gg(W, []))
STAR_IN_GG([], W) → =_IN_GG(W, [])
STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
STAR_IN_GG(U, W) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U5_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U3_GG(U, W, app_out_gag(U, V, W)) → U4_GG(U, W, star_in_gg(U, V))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2) = star_in_gg(x1, x2)
[] = []
U1_gg(x1, x2) = U1_gg(x2)
true_in_ = true_in_
true_out_ = true_out_
star_out_gg(x1, x2) = star_out_gg
U2_gg(x1, x2) = U2_gg(x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U3_gg(x1, x2, x3) = U3_gg(x1, x3)
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x2)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x5)
U4_gg(x1, x2, x3) = U4_gg(x3)
STAR_IN_GG(x1, x2) = STAR_IN_GG(x1, x2)
U1_GG(x1, x2) = U1_GG(x2)
TRUE_IN_ = TRUE_IN_
U2_GG(x1, x2) = U2_GG(x2)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, x3)
APP_IN_GAG(x1, x2, x3) = APP_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x5)
U4_GG(x1, x2, x3) = U4_GG(x3)

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

(8) Obligation:

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

STAR_IN_GG(U, []) → U1_GG(U, true_in_)
STAR_IN_GG(U, []) → TRUE_IN_
STAR_IN_GG([], W) → U2_GG(W, =_in_gg(W, []))
STAR_IN_GG([], W) → =_IN_GG(W, [])
STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
STAR_IN_GG(U, W) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U5_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U3_GG(U, W, app_out_gag(U, V, W)) → U4_GG(U, W, star_in_gg(U, V))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2) = star_in_gg(x1, x2)
[] = []
U1_gg(x1, x2) = U1_gg(x2)
true_in_ = true_in_
true_out_ = true_out_
star_out_gg(x1, x2) = star_out_gg
U2_gg(x1, x2) = U2_gg(x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U3_gg(x1, x2, x3) = U3_gg(x1, x3)
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x2)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x5)
U4_gg(x1, x2, x3) = U4_gg(x3)
APP_IN_GAG(x1, x2, x3) = APP_IN_GAG(x1, x3)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

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

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

APP_IN_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)

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

(16) 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_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)
The graph contains the following edges 1 > 1, 2 > 2

(17) TRUE

(18) Obligation:

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

STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2) = star_in_gg(x1, x2)
[] = []
U1_gg(x1, x2) = U1_gg(x2)
true_in_ = true_in_
true_out_ = true_out_
star_out_gg(x1, x2) = star_out_gg
U2_gg(x1, x2) = U2_gg(x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U3_gg(x1, x2, x3) = U3_gg(x1, x3)
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x2)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x5)
U4_gg(x1, x2, x3) = U4_gg(x3)
STAR_IN_GG(x1, x2) = STAR_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, 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:

STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The argument filtering Pi contains the following mapping:
[] = []
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x2)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x5)
STAR_IN_GG(x1, x2) = STAR_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, 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:

STAR_IN_GG(U, W) → U3_GG(U, app_in_gag(U, W))
U3_GG(U, app_out_gag(V)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(23) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule STAR_IN_GG(U, W) → U3_GG(U, app_in_gag(U, W)) at position [1] we obtained the following new rules [LPAR04]:

STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), U5_gag(app_in_gag(x1, x2)))

(24) Obligation:

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

U3_GG(U, app_out_gag(V)) → STAR_IN_GG(U, V)
STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), U5_gag(app_in_gag(x1, x2)))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(25) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GG(U, app_out_gag(V)) → STAR_IN_GG(U, V) we obtained the following new rules [LPAR04]:

U3_GG([], app_out_gag(z0)) → STAR_IN_GG([], z0)
U3_GG(.(z0, z1), app_out_gag(x1)) → STAR_IN_GG(.(z0, z1), x1)

(26) Obligation:

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

STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), U5_gag(app_in_gag(x1, x2)))
U3_GG([], app_out_gag(z0)) → STAR_IN_GG([], z0)
U3_GG(.(z0, z1), app_out_gag(x1)) → STAR_IN_GG(.(z0, z1), x1)

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(28) Complex Obligation (AND)

(29) Obligation:

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

U3_GG(.(z0, z1), app_out_gag(x1)) → STAR_IN_GG(.(z0, z1), x1)
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), U5_gag(app_in_gag(x1, x2)))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(30) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U3_GG(.(z0, z1), app_out_gag(x1)) → STAR_IN_GG(.(z0, z1), x1) we obtained the following new rules [LPAR04]:

U3_GG(.(x0, x1), app_out_gag(.(y_2, y_3))) → STAR_IN_GG(.(x0, x1), .(y_2, y_3))

(31) Obligation:

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

STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), U5_gag(app_in_gag(x1, x2)))
U3_GG(.(x0, x1), app_out_gag(.(y_2, y_3))) → STAR_IN_GG(.(x0, x1), .(y_2, y_3))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U3_GG(.(x0, x1), app_out_gag(.(y_2, y_3))) → STAR_IN_GG(.(x0, x1), .(y_2, y_3))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(STAR_IN_GG(x₁, x₂)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(.(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(U3_GG(x₁, x₂)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(U5_gag(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(app_in_gag(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 1 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(app_out_gag(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL([]) =
/ 1 \

\ 1 /

The following usable rules [FROCOS05] were oriented:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

(33) Obligation:

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

STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), U5_gag(app_in_gag(x1, x2)))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(35) TRUE

(36) Obligation:

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

U3_GG([], app_out_gag(z0)) → STAR_IN_GG([], z0)
STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(app_in_gag(L, N))
U5_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(37) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(38) Obligation:

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

U3_GG([], app_out_gag(z0)) → STAR_IN_GG([], z0)
STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))

R is empty.
The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0)

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

(39) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

app_in_gag(x0, x1)
U5_gag(x0)

(40) Obligation:

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

U3_GG([], app_out_gag(z0)) → STAR_IN_GG([], z0)
STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))

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

(41) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = STAR_IN_GG([], x0) evaluates to t =STAR_IN_GG([], x0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

STAR_IN_GG([], x0) → U3_GG([], app_out_gag(x0))
with rule STAR_IN_GG([], x0') → U3_GG([], app_out_gag(x0')) at position [] and matcher [x0' / x0]

U3_GG([], app_out_gag(x0)) → STAR_IN_GG([], x0)
with rule U3_GG([], app_out_gag(z0)) → STAR_IN_GG([], z0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(42) FALSE

(43) PrologToPiTRSProof (SOUND transformation)

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(44) Obligation:

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

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

(45) 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:

STAR_IN_GG(U, []) → U1_GG(U, true_in_)
STAR_IN_GG(U, []) → TRUE_IN_
STAR_IN_GG([], W) → U2_GG(W, =_in_gg(W, []))
STAR_IN_GG([], W) → =_IN_GG(W, [])
STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
STAR_IN_GG(U, W) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U5_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U3_GG(U, W, app_out_gag(U, V, W)) → U4_GG(U, W, star_in_gg(U, V))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2) = star_in_gg(x1, x2)
[] = []
U1_gg(x1, x2) = U1_gg(x1, x2)
true_in_ = true_in_
true_out_ = true_out_
star_out_gg(x1, x2) = star_out_gg(x1, x2)
U2_gg(x1, x2) = U2_gg(x1, x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg(x1, x2)
U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3)
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x1, x2, x4, x5)
U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3)
STAR_IN_GG(x1, x2) = STAR_IN_GG(x1, x2)
U1_GG(x1, x2) = U1_GG(x1, x2)
TRUE_IN_ = TRUE_IN_
U2_GG(x1, x2) = U2_GG(x1, x2)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, x2, x3)
APP_IN_GAG(x1, x2, x3) = APP_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x1, x2, x4, x5)
U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3)

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

(46) Obligation:

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

STAR_IN_GG(U, []) → U1_GG(U, true_in_)
STAR_IN_GG(U, []) → TRUE_IN_
STAR_IN_GG([], W) → U2_GG(W, =_in_gg(W, []))
STAR_IN_GG([], W) → =_IN_GG(W, [])
STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
STAR_IN_GG(U, W) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U5_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U3_GG(U, W, app_out_gag(U, V, W)) → U4_GG(U, W, star_in_gg(U, V))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(48) Complex Obligation (AND)

(49) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

(50) UsableRulesProof (EQUIVALENT transformation)

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

(51) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

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

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

(52) PiDPToQDPProof (SOUND transformation)

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

(53) Obligation:

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

APP_IN_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)

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

(54) 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_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)
The graph contains the following edges 1 > 1, 2 > 2

(55) TRUE

(56) Obligation:

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

STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

star_in_gg(U, []) → U1_gg(U, true_in_)
true_in_ → true_out_
U1_gg(U, true_out_) → star_out_gg(U, [])
star_in_gg([], W) → U2_gg(W, =_in_gg(W, []))
=_in_gg(X, X) → =_out_gg(X, X)
U2_gg(W, =_out_gg(W, [])) → star_out_gg([], W)
star_in_gg(U, W) → U3_gg(U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U3_gg(U, W, app_out_gag(U, V, W)) → U4_gg(U, W, star_in_gg(U, V))
U4_gg(U, W, star_out_gg(U, V)) → star_out_gg(U, W)

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2) = star_in_gg(x1, x2)
[] = []
U1_gg(x1, x2) = U1_gg(x1, x2)
true_in_ = true_in_
true_out_ = true_out_
star_out_gg(x1, x2) = star_out_gg(x1, x2)
U2_gg(x1, x2) = U2_gg(x1, x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg(x1, x2)
U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3)
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x1, x2, x4, x5)
U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3)
STAR_IN_GG(x1, x2) = STAR_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, x2, x3)

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

(57) UsableRulesProof (EQUIVALENT transformation)

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

(58) Obligation:

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

STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, V, W))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U5_gag(X, L, M, N, app_in_gag(L, M, N))
U5_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The argument filtering Pi contains the following mapping:
[] = []
app_in_gag(x1, x2, x3) = app_in_gag(x1, x3)
app_out_gag(x1, x2, x3) = app_out_gag(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x1, x2, x4, x5)
STAR_IN_GG(x1, x2) = STAR_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, x2, x3)

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

(59) PiDPToQDPProof (SOUND transformation)

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

(60) Obligation:

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

STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, W))
U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(61) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule STAR_IN_GG(U, W) → U3_GG(U, W, app_in_gag(U, W)) at position [2] we obtained the following new rules [LPAR04]:

STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), .(x0, x2), U5_gag(x0, x1, x2, app_in_gag(x1, x2)))

(62) Obligation:

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

U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V)
STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), .(x0, x2), U5_gag(x0, x1, x2, app_in_gag(x1, x2)))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(63) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GG(U, W, app_out_gag(U, V, W)) → STAR_IN_GG(U, V) we obtained the following new rules [LPAR04]:

U3_GG([], z0, app_out_gag([], z0, z0)) → STAR_IN_GG([], z0)
U3_GG(.(z0, z1), .(z0, z2), app_out_gag(.(z0, z1), x2, .(z0, z2))) → STAR_IN_GG(.(z0, z1), x2)

(64) Obligation:

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

STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), .(x0, x2), U5_gag(x0, x1, x2, app_in_gag(x1, x2)))
U3_GG([], z0, app_out_gag([], z0, z0)) → STAR_IN_GG([], z0)
U3_GG(.(z0, z1), .(z0, z2), app_out_gag(.(z0, z1), x2, .(z0, z2))) → STAR_IN_GG(.(z0, z1), x2)

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(65) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(66) Complex Obligation (AND)

(67) Obligation:

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

U3_GG(.(z0, z1), .(z0, z2), app_out_gag(.(z0, z1), x2, .(z0, z2))) → STAR_IN_GG(.(z0, z1), x2)
STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), .(x0, x2), U5_gag(x0, x1, x2, app_in_gag(x1, x2)))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(68) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U3_GG(.(z0, z1), .(z0, z2), app_out_gag(.(z0, z1), x2, .(z0, z2))) → STAR_IN_GG(.(z0, z1), x2) we obtained the following new rules [LPAR04]:

U3_GG(.(x0, x1), .(x0, x2), app_out_gag(.(x0, x1), .(y_2, y_3), .(x0, x2))) → STAR_IN_GG(.(x0, x1), .(y_2, y_3))

(69) Obligation:

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

STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), .(x0, x2), U5_gag(x0, x1, x2, app_in_gag(x1, x2)))
U3_GG(.(x0, x1), .(x0, x2), app_out_gag(.(x0, x1), .(y_2, y_3), .(x0, x2))) → STAR_IN_GG(.(x0, x1), .(y_2, y_3))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U3_GG(.(x0, x1), .(x0, x2), app_out_gag(.(x0, x1), .(y_2, y_3), .(x0, x2))) → STAR_IN_GG(.(x0, x1), .(y_2, y_3))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(STAR_IN_GG(x₁, x₂)) = x₂
POL(U3_GG(x₁, x₂, x₃)) = 1 + x₃
POL(U5_gag(x₁, x₂, x₃, x₄)) = x₄
POL([]) = 0
POL(app_in_gag(x₁, x₂)) = x₂
POL(app_out_gag(x₁, x₂, x₃)) = x₂

The following usable rules [FROCOS05] were oriented:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

(71) Obligation:

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

STAR_IN_GG(.(x0, x1), .(x0, x2)) → U3_GG(.(x0, x1), .(x0, x2), U5_gag(x0, x1, x2, app_in_gag(x1, x2)))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(72) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(73) TRUE

(74) Obligation:

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

U3_GG([], z0, app_out_gag([], z0, z0)) → STAR_IN_GG([], z0)
STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U5_gag(X, L, N, app_in_gag(L, N))
U5_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(75) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(76) Obligation:

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

U3_GG([], z0, app_out_gag([], z0, z0)) → STAR_IN_GG([], z0)
STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))

R is empty.
The set Q consists of the following terms:

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

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

(77) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

app_in_gag(x0, x1)
U5_gag(x0, x1, x2, x3)

(78) Obligation:

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

U3_GG([], z0, app_out_gag([], z0, z0)) → STAR_IN_GG([], z0)
STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))

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

(79) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

STAR_IN_GG([], x0) → U3_GG([], x0, app_out_gag([], x0, x0))
with rule STAR_IN_GG([], x0') → U3_GG([], x0', app_out_gag([], x0', x0')) at position [] and matcher [x0' / x0]

U3_GG([], x0, app_out_gag([], x0, x0)) → STAR_IN_GG([], x0)
with rule U3_GG([], z0, app_out_gag([], z0, z0)) → STAR_IN_GG([], z0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.