(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

star1(T3, []).
star1(T5, T6) :- app13(T5, X11, T6).
star1(T5, T6) :- ','(app13(T5, T7, T6), star1(T5, T7)).
app19([], T11, T11).
app19(.(T12, T13), X35, .(T12, T14)) :- app19(T13, X35, T14).
app13(.(T8, []), T11, .(T8, T11)).
app13(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) :- app19(T13, X35, T14).

Queries:

star1(g,g).

(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:
star1_in: (b,b)
app13_in: (b,f,b)
app19_in: (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)

(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:

STAR1_IN_GG(T5, T6) → U1_GG(T5, T6, app13_in_gag(T5, X11, T6))
STAR1_IN_GG(T5, T6) → APP13_IN_GAG(T5, X11, T6)
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_GAG(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → APP19_IN_GAG(T13, X35, T14)
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → U4_GAG(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)
STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → U3_GG(T5, T6, star1_in_gg(T5, T7))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
APP13_IN_GAG(x1, x2, x3)  =  APP13_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x5)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)
U3_GG(x1, x2, x3)  =  U3_GG(x3)

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

(6) Obligation:

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

STAR1_IN_GG(T5, T6) → U1_GG(T5, T6, app13_in_gag(T5, X11, T6))
STAR1_IN_GG(T5, T6) → APP13_IN_GAG(T5, X11, T6)
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_GAG(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → APP19_IN_GAG(T13, X35, T14)
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → U4_GAG(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)
STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → U3_GG(T5, T6, star1_in_gg(T5, T7))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
APP13_IN_GAG(x1, x2, x3)  =  APP13_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x5)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)
U3_GG(x1, x2, x3)  =  U3_GG(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP19_IN_GAG(.(T12, T13), .(T12, T14)) → APP19_IN_GAG(T13, T14)

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

(14) 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:

  • APP19_IN_GAG(.(T12, T13), .(T12, T14)) → APP19_IN_GAG(T13, T14)
    The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))

The argument filtering Pi contains the following mapping:
[]  =  []
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

STAR1_IN_GG(T5, T6) → U2_GG(T5, app13_in_gag(T5, T6))
U2_GG(T5, app13_out_gag(T7)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

app13_in_gag(.(T8, []), .(T8, T11)) → app13_out_gag(T11)
app13_in_gag(.(T8, .(T12, T13)), .(T8, .(T12, T14))) → U5_gag(app19_in_gag(T13, T14))
U5_gag(app19_out_gag(X35)) → app13_out_gag(X35)
app19_in_gag([], T11) → app19_out_gag(T11)
app19_in_gag(.(T12, T13), .(T12, T14)) → U4_gag(app19_in_gag(T13, T14))
U4_gag(app19_out_gag(X35)) → app19_out_gag(X35)

The set Q consists of the following terms:

app13_in_gag(x0, x1)
U5_gag(x0)
app19_in_gag(x0, x1)
U4_gag(x0)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2_GG(T5, app13_out_gag(T7)) → STAR1_IN_GG(T5, T7)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(STAR1_IN_GG(x1, x2)) = x2   
POL(U2_GG(x1, x2)) = x2   
POL(U4_gag(x1)) = 1 + x1   
POL(U5_gag(x1)) = 1 + x1   
POL([]) = 0   
POL(app13_in_gag(x1, x2)) = x2   
POL(app13_out_gag(x1)) = 1 + x1   
POL(app19_in_gag(x1, x2)) = 1 + x2   
POL(app19_out_gag(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

app13_in_gag(.(T8, []), .(T8, T11)) → app13_out_gag(T11)
app13_in_gag(.(T8, .(T12, T13)), .(T8, .(T12, T14))) → U5_gag(app19_in_gag(T13, T14))
app19_in_gag([], T11) → app19_out_gag(T11)
app19_in_gag(.(T12, T13), .(T12, T14)) → U4_gag(app19_in_gag(T13, T14))
U5_gag(app19_out_gag(X35)) → app13_out_gag(X35)
U4_gag(app19_out_gag(X35)) → app19_out_gag(X35)

(22) Obligation:

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

STAR1_IN_GG(T5, T6) → U2_GG(T5, app13_in_gag(T5, T6))

The TRS R consists of the following rules:

app13_in_gag(.(T8, []), .(T8, T11)) → app13_out_gag(T11)
app13_in_gag(.(T8, .(T12, T13)), .(T8, .(T12, T14))) → U5_gag(app19_in_gag(T13, T14))
U5_gag(app19_out_gag(X35)) → app13_out_gag(X35)
app19_in_gag([], T11) → app19_out_gag(T11)
app19_in_gag(.(T12, T13), .(T12, T14)) → U4_gag(app19_in_gag(T13, T14))
U4_gag(app19_out_gag(X35)) → app19_out_gag(X35)

The set Q consists of the following terms:

app13_in_gag(x0, x1)
U5_gag(x0)
app19_in_gag(x0, x1)
U4_gag(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE