Left Termination of the query pattern lessleaves_in_2(g, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

lessleaves(nil, cons(X1, X2)).
lessleaves(cons(U, V), cons(W, Z)) :- ','(app(U, V, U1), ','(app(W, Z, W1), lessleaves(U1, W1))).
app(nil, Y, Z) :- ','(!, eq(Y, Z)).
app(X, Y, cons(U, Z)) :- ','(head(X, U), ','(tail(X, V), app(V, Y, Z))).
head([], X3).
head(.(X, X4), X).
tail([], []).
tail(.(X5, X), X).
eq(X, X).

Queries:

lessleaves(g,g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

lessleaves(nil, cons(X1, X2)).
lessleaves(cons(U, V), cons(W, Z)) :- ','(app(U, V, U1), ','(app(W, Z, W1), lessleaves(U1, W1))).
app(nil, Y, Z) :- eq(Y, Z).
app(X, Y, cons(U, Z)) :- ','(head(X, U), ','(tail(X, V), app(V, Y, Z))).
head([], X3).
head(.(X, X4), X).
tail([], []).
tail(.(X5, X), X).
eq(X, X).

Queries:

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

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)

(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:

LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → U1_GG(U, V, W, Z, app_in_gga(U, V, U1))
LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → APP_IN_GGA(U, V, U1)
APP_IN_GGA(nil, Y, Z) → U4_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA(nil, Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
APP_IN_GGA(X, Y, cons(U, Z)) → HEAD_IN_GA(X, U)
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → TAIL_IN_GA(X, V)
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → U7_GGA(X, Y, U, Z, app_in_gga(V, Y, Z))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → U2_GG(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → APP_IN_GGA(W, Z, W1)
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_GG(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → APP_IN_AAA(U, V, U1)
APP_IN_AAA(nil, Y, Z) → U4_AAA(Y, Z, eq_in_aa(Y, Z))
APP_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
APP_IN_AAA(X, Y, cons(U, Z)) → HEAD_IN_AA(X, U)
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → TAIL_IN_AA(X, V)
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → U7_AAA(X, Y, U, Z, app_in_aaa(V, Y, Z))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → APP_IN_AAA(W, Z, W1)
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_AA(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
LESSLEAVES_IN_GG(x1, x2)  =  LESSLEAVES_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3)  =  U4_GGA(x1, x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x5)
U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x1, x2, x3, x4, x6)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

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

(6) Obligation:

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

LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → U1_GG(U, V, W, Z, app_in_gga(U, V, U1))
LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → APP_IN_GGA(U, V, U1)
APP_IN_GGA(nil, Y, Z) → U4_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA(nil, Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
APP_IN_GGA(X, Y, cons(U, Z)) → HEAD_IN_GA(X, U)
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → TAIL_IN_GA(X, V)
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → U7_GGA(X, Y, U, Z, app_in_gga(V, Y, Z))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → U2_GG(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → APP_IN_GGA(W, Z, W1)
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_GG(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → APP_IN_AAA(U, V, U1)
APP_IN_AAA(nil, Y, Z) → U4_AAA(Y, Z, eq_in_aa(Y, Z))
APP_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
APP_IN_AAA(X, Y, cons(U, Z)) → HEAD_IN_AA(X, U)
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → TAIL_IN_AA(X, V)
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → U7_AAA(X, Y, U, Z, app_in_aaa(V, Y, Z))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → APP_IN_AAA(W, Z, W1)
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_AA(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
LESSLEAVES_IN_GG(x1, x2)  =  LESSLEAVES_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3)  =  U4_GGA(x1, x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x5)
U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x1, x2, x3, x4, x6)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 19 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)

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:

APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)

The TRS R consists of the following rules:

head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
.(x1, x2)  =  .(x1, x2)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)

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:

APP_IN_AAA → U5_AAA(head_in_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(14) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule APP_IN_AAA → U5_AAA(head_in_aa) at position [0] we obtained the following new rules [LPAR04]:

APP_IN_AAA → U5_AAA(head_out_aa)

(15) Obligation:

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

U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(16) 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.

(17) Obligation:

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

U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(18) 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].

head_in_aa

(19) Obligation:

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

U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(20) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_AAA(head_out_aa) → U6_AAA(tail_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

(21) Obligation:

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

U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(22) 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.

(23) Obligation:

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

U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

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

tail_in_aa

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

(24) 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].

tail_in_aa

(25) Obligation:

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

U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

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

(26) 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 = APP_IN_AAA evaluates to t =APP_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

APP_IN_AAA → U5_AAA(head_out_aa)
with rule APP_IN_AAA → U5_AAA(head_out_aa) at position [] and matcher [ ]

U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)
with rule U5_AAA(head_out_aa) → U6_AAA(tail_out_aa) at position [] and matcher [ ]

U6_AAA(tail_out_aa) → APP_IN_AAA
with rule U6_AAA(tail_out_aa) → APP_IN_AAA

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.

(27) FALSE

(28) Obligation:

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

U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))

The TRS R consists of the following rules:

app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
eq_in_aa(X, X) → eq_out_aa(X, X)
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

U1_AA(app_out_aaa) → U2_AA(app_in_aaa)
U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(33) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(app_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_in_aa))
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa))

(34) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_in_aa))
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(35) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa))

(36) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa))
U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(37) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))

(38) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa))
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(39) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(app_out_aaa)

(40) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(41) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(app_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_in_aa))
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa))

(42) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_in_aa))
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(43) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa))

(44) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa))
LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(45) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_out_aa))

(46) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa))
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(47) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(app_out_aaa)

(48) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_out_aa))
LESSLEAVES_IN_AA → U1_AA(app_out_aaa)

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(49) 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 = LESSLEAVES_IN_AA evaluates to t =LESSLEAVES_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

LESSLEAVES_IN_AA → U1_AA(app_out_aaa)
with rule LESSLEAVES_IN_AA → U1_AA(app_out_aaa) at position [] and matcher [ ]

U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
with rule U1_AA(app_out_aaa) → U2_AA(app_out_aaa) at position [] and matcher [ ]

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
with rule U2_AA(app_out_aaa) → LESSLEAVES_IN_AA

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.

(50) FALSE

(51) Obligation:

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

APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x1, x2, x3, x4, x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x5)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)

The TRS R consists of the following rules:

head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
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)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x5)

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

(54) PiDPToQDPProof (SOUND transformation)

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

(55) Obligation:

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

APP_IN_GGA(X, Y) → U5_GGA(X, Y, head_in_ga(X))
U5_GGA(X, Y, head_out_ga(X)) → U6_GGA(X, Y, tail_in_ga(X))
U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X, X4)) → head_out_ga(.(X, X4))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, X)) → tail_out_ga(.(X5, X), X)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(56) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule APP_IN_GGA(X, Y) → U5_GGA(X, Y, head_in_ga(X)) at position [2] we obtained the following new rules [LPAR04]:

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))

(57) Obligation:

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

U5_GGA(X, Y, head_out_ga(X)) → U6_GGA(X, Y, tail_in_ga(X))
U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X, X4)) → head_out_ga(.(X, X4))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, X)) → tail_out_ga(.(X5, X), X)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(58) 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.

(59) Obligation:

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

U5_GGA(X, Y, head_out_ga(X)) → U6_GGA(X, Y, tail_in_ga(X))
U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, X)) → tail_out_ga(.(X5, X), X)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(60) 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].

head_in_ga(x0)

(61) Obligation:

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

U5_GGA(X, Y, head_out_ga(X)) → U6_GGA(X, Y, tail_in_ga(X))
U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, X)) → tail_out_ga(.(X5, X), X)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(62) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_GGA(X, Y, head_out_ga(X)) → U6_GGA(X, Y, tail_in_ga(X)) at position [2] we obtained the following new rules [LPAR04]:

U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1))) → U6_GGA(.(x0, x1), y1, tail_out_ga(.(x0, x1), x1))

(63) Obligation:

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

U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1))) → U6_GGA(.(x0, x1), y1, tail_out_ga(.(x0, x1), x1))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(X5, X)) → tail_out_ga(.(X5, X), X)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(64) 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.

(65) Obligation:

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

U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1))) → U6_GGA(.(x0, x1), y1, tail_out_ga(.(x0, x1), x1))

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

tail_in_ga(x0)

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

(66) 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].

tail_in_ga(x0)

(67) Obligation:

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

U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1))) → U6_GGA(.(x0, x1), y1, tail_out_ga(.(x0, x1), x1))

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1))) → U6_GGA(.(x0, x1), y1, tail_out_ga(.(x0, x1), x1))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(APP_IN_GGA(x₁, x₂)) = x₁   
POL(U5_GGA(x₁, x₂, x₃)) = x₃   
POL(U6_GGA(x₁, x₂, x₃)) = x₃   
POL([]) = 0   
POL(head_out_ga(x₁)) = x₁   
POL(tail_out_ga(x₁, x₂)) = x₂   

The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

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

U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0, x1), y1) → U5_GGA(.(x0, x1), y1, head_out_ga(.(x0, x1)))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))

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

(70) DependencyGraphProof (EQUIVALENT transformation)

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

(71) Obligation:

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

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y)

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

(72) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y) we obtained the following new rules [LPAR04]:

U6_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)

(73) Obligation:

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

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U6_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)

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

(74) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GGA(X, Y, tail_out_ga(X, V)) → APP_IN_GGA(V, Y) we obtained the following new rules [LPAR04]:

U6_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)

(75) Obligation:

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

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
U6_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)

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

(76) 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 = U5_GGA([], y1, head_out_ga([])) evaluates to t =U5_GGA([], y1, head_out_ga([]))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U5_GGA([], y1, head_out_ga([])) → U6_GGA([], y1, tail_out_ga([], []))
with rule U5_GGA([], y1', head_out_ga([])) → U6_GGA([], y1', tail_out_ga([], [])) at position [] and matcher [y1' / y1]

U6_GGA([], y1, tail_out_ga([], [])) → APP_IN_GGA([], y1)
with rule U6_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0) at position [] and matcher [z0 / y1]

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))
with rule APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga([]))

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.

(77) FALSE

(78) PrologToPiTRSProof (SOUND transformation)

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

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(79) Obligation:

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

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)

(80) 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:

LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → U1_GG(U, V, W, Z, app_in_gga(U, V, U1))
LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → APP_IN_GGA(U, V, U1)
APP_IN_GGA(nil, Y, Z) → U4_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA(nil, Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
APP_IN_GGA(X, Y, cons(U, Z)) → HEAD_IN_GA(X, U)
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → TAIL_IN_GA(X, V)
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → U7_GGA(X, Y, U, Z, app_in_gga(V, Y, Z))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → U2_GG(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → APP_IN_GGA(W, Z, W1)
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_GG(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → APP_IN_AAA(U, V, U1)
APP_IN_AAA(nil, Y, Z) → U4_AAA(Y, Z, eq_in_aa(Y, Z))
APP_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
APP_IN_AAA(X, Y, cons(U, Z)) → HEAD_IN_AA(X, U)
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → TAIL_IN_AA(X, V)
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → U7_AAA(X, Y, U, Z, app_in_aaa(V, Y, Z))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → APP_IN_AAA(W, Z, W1)
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_AA(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
LESSLEAVES_IN_GG(x1, x2)  =  LESSLEAVES_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3)  =  U4_GGA(x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x2, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x5)
U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x6)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

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

(81) Obligation:

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

LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → U1_GG(U, V, W, Z, app_in_gga(U, V, U1))
LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) → APP_IN_GGA(U, V, U1)
APP_IN_GGA(nil, Y, Z) → U4_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA(nil, Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
APP_IN_GGA(X, Y, cons(U, Z)) → HEAD_IN_GA(X, U)
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → TAIL_IN_GA(X, V)
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → U7_GGA(X, Y, U, Z, app_in_gga(V, Y, Z))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → U2_GG(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U1_GG(U, V, W, Z, app_out_gga(U, V, U1)) → APP_IN_GGA(W, Z, W1)
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_GG(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_GG(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → APP_IN_AAA(U, V, U1)
APP_IN_AAA(nil, Y, Z) → U4_AAA(Y, Z, eq_in_aa(Y, Z))
APP_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
APP_IN_AAA(X, Y, cons(U, Z)) → HEAD_IN_AA(X, U)
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → TAIL_IN_AA(X, V)
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → U7_AAA(X, Y, U, Z, app_in_aaa(V, Y, Z))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → APP_IN_AAA(W, Z, W1)
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_AA(U, V, W, Z, lessleaves_in_aa(U1, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
LESSLEAVES_IN_GG(x1, x2)  =  LESSLEAVES_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3)  =  U4_GGA(x3)
EQ_IN_GA(x1, x2)  =  EQ_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x2, x5)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x5)
U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x6)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

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

(82) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 19 less nodes.

(83) Complex Obligation (AND)

(84) Obligation:

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

APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)

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

(85) UsableRulesProof (EQUIVALENT transformation)

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

(86) Obligation:

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

APP_IN_AAA(X, Y, cons(U, Z)) → U5_AAA(X, Y, U, Z, head_in_aa(X, U))
U5_AAA(X, Y, U, Z, head_out_aa(X, U)) → U6_AAA(X, Y, U, Z, tail_in_aa(X, V))
U6_AAA(X, Y, U, Z, tail_out_aa(X, V)) → APP_IN_AAA(V, Y, Z)

The TRS R consists of the following rules:

head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
.(x1, x2)  =  .(x1, x2)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)

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

(87) PiDPToQDPProof (SOUND transformation)

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

(88) Obligation:

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

APP_IN_AAA → U5_AAA(head_in_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(89) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule APP_IN_AAA → U5_AAA(head_in_aa) at position [0] we obtained the following new rules [LPAR04]:

APP_IN_AAA → U5_AAA(head_out_aa)

(90) Obligation:

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

U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(91) 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.

(92) Obligation:

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

U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(93) 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].

head_in_aa

(94) Obligation:

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

U5_AAA(head_out_aa) → U6_AAA(tail_in_aa)
U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(95) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_AAA(head_out_aa) → U6_AAA(tail_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

(96) Obligation:

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

U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(97) 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.

(98) Obligation:

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

U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

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

tail_in_aa

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

(99) 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].

tail_in_aa

(100) Obligation:

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

U6_AAA(tail_out_aa) → APP_IN_AAA
APP_IN_AAA → U5_AAA(head_out_aa)
U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)

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

(101) 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 = APP_IN_AAA evaluates to t =APP_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

APP_IN_AAA → U5_AAA(head_out_aa)
with rule APP_IN_AAA → U5_AAA(head_out_aa) at position [] and matcher [ ]

U5_AAA(head_out_aa) → U6_AAA(tail_out_aa)
with rule U5_AAA(head_out_aa) → U6_AAA(tail_out_aa) at position [] and matcher [ ]

U6_AAA(tail_out_aa) → APP_IN_AAA
with rule U6_AAA(tail_out_aa) → APP_IN_AAA

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.

(102) FALSE

(103) Obligation:

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

U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(104) UsableRulesProof (EQUIVALENT transformation)

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

(105) Obligation:

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

U1_AA(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_AA(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_AA(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → LESSLEAVES_IN_AA(U1, W1)
LESSLEAVES_IN_AA(cons(U, V), cons(W, Z)) → U1_AA(U, V, W, Z, app_in_aaa(U, V, U1))

The TRS R consists of the following rules:

app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
eq_in_aa(X, X) → eq_out_aa(X, X)
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
LESSLEAVES_IN_AA(x1, x2)  =  LESSLEAVES_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(106) PiDPToQDPProof (SOUND transformation)

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

(107) Obligation:

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

U1_AA(app_out_aaa) → U2_AA(app_in_aaa)
U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(108) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(app_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_in_aa))
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa))

(109) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_in_aa))
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(110) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa))

(111) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa))
U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(112) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))

(113) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa))
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(114) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U1_AA(app_out_aaa) → U2_AA(U4_aaa(eq_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

U1_AA(app_out_aaa) → U2_AA(app_out_aaa)

(115) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
LESSLEAVES_IN_AA → U1_AA(app_in_aaa)
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(116) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(app_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_in_aa))
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa))

(117) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_in_aa))
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(118) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa))

(119) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa))
LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(120) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_out_aa))

(121) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa))
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_out_aa))

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(122) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule LESSLEAVES_IN_AA → U1_AA(U4_aaa(eq_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

LESSLEAVES_IN_AA → U1_AA(app_out_aaa)

(123) Obligation:

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

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
U1_AA(app_out_aaa) → U2_AA(U5_aaa(head_out_aa))
U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
LESSLEAVES_IN_AA → U1_AA(U5_aaa(head_out_aa))
LESSLEAVES_IN_AA → U1_AA(app_out_aaa)

The TRS R consists of the following rules:

app_in_aaa → U4_aaa(eq_in_aa)
app_in_aaa → U5_aaa(head_in_aa)
U4_aaa(eq_out_aa) → app_out_aaa
U5_aaa(head_out_aa) → U6_aaa(tail_in_aa)
eq_in_aa → eq_out_aa
head_in_aa → head_out_aa
U6_aaa(tail_out_aa) → U7_aaa(app_in_aaa)
tail_in_aa → tail_out_aa
U7_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U4_aaa(x0)
U5_aaa(x0)
eq_in_aa
head_in_aa
U6_aaa(x0)
tail_in_aa
U7_aaa(x0)

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

(124) 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 = LESSLEAVES_IN_AA evaluates to t =LESSLEAVES_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

LESSLEAVES_IN_AA → U1_AA(app_out_aaa)
with rule LESSLEAVES_IN_AA → U1_AA(app_out_aaa) at position [] and matcher [ ]

U1_AA(app_out_aaa) → U2_AA(app_out_aaa)
with rule U1_AA(app_out_aaa) → U2_AA(app_out_aaa) at position [] and matcher [ ]

U2_AA(app_out_aaa) → LESSLEAVES_IN_AA
with rule U2_AA(app_out_aaa) → LESSLEAVES_IN_AA

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.

(125) FALSE

(126) Obligation:

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

APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)

The TRS R consists of the following rules:

lessleaves_in_gg(nil, cons(X1, X2)) → lessleaves_out_gg(nil, cons(X1, X2))
lessleaves_in_gg(cons(U, V), cons(W, Z)) → U1_gg(U, V, W, Z, app_in_gga(U, V, U1))
app_in_gga(nil, Y, Z) → U4_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U4_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga(nil, Y, Z)
app_in_gga(X, Y, cons(U, Z)) → U5_gga(X, Y, U, Z, head_in_ga(X, U))
head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
U5_gga(X, Y, U, Z, head_out_ga(X, U)) → U6_gga(X, Y, U, Z, tail_in_ga(X, V))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)
U6_gga(X, Y, U, Z, tail_out_ga(X, V)) → U7_gga(X, Y, U, Z, app_in_gga(V, Y, Z))
U7_gga(X, Y, U, Z, app_out_gga(V, Y, Z)) → app_out_gga(X, Y, cons(U, Z))
U1_gg(U, V, W, Z, app_out_gga(U, V, U1)) → U2_gg(U, V, W, Z, U1, app_in_gga(W, Z, W1))
U2_gg(U, V, W, Z, U1, app_out_gga(W, Z, W1)) → U3_gg(U, V, W, Z, lessleaves_in_aa(U1, W1))
lessleaves_in_aa(nil, cons(X1, X2)) → lessleaves_out_aa(nil, cons(X1, X2))
lessleaves_in_aa(cons(U, V), cons(W, Z)) → U1_aa(U, V, W, Z, app_in_aaa(U, V, U1))
app_in_aaa(nil, Y, Z) → U4_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U4_aaa(Y, Z, eq_out_aa(Y, Z)) → app_out_aaa(nil, Y, Z)
app_in_aaa(X, Y, cons(U, Z)) → U5_aaa(X, Y, U, Z, head_in_aa(X, U))
head_in_aa([], X3) → head_out_aa([], X3)
head_in_aa(.(X, X4), X) → head_out_aa(.(X, X4), X)
U5_aaa(X, Y, U, Z, head_out_aa(X, U)) → U6_aaa(X, Y, U, Z, tail_in_aa(X, V))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X5, X), X) → tail_out_aa(.(X5, X), X)
U6_aaa(X, Y, U, Z, tail_out_aa(X, V)) → U7_aaa(X, Y, U, Z, app_in_aaa(V, Y, Z))
U7_aaa(X, Y, U, Z, app_out_aaa(V, Y, Z)) → app_out_aaa(X, Y, cons(U, Z))
U1_aa(U, V, W, Z, app_out_aaa(U, V, U1)) → U2_aa(U, V, W, Z, U1, app_in_aaa(W, Z, W1))
U2_aa(U, V, W, Z, U1, app_out_aaa(W, Z, W1)) → U3_aa(U, V, W, Z, lessleaves_in_aa(U1, W1))
U3_aa(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_aa(cons(U, V), cons(W, Z))
U3_gg(U, V, W, Z, lessleaves_out_aa(U1, W1)) → lessleaves_out_gg(cons(U, V), cons(W, Z))

The argument filtering Pi contains the following mapping:
lessleaves_in_gg(x1, x2)  =  lessleaves_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves_out_gg(x1, x2)  =  lessleaves_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U4_gga(x1, x2, x3)  =  U4_gga(x3)
eq_in_ga(x1, x2)  =  eq_in_ga(x1)
eq_out_ga(x1, x2)  =  eq_out_ga(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x5)
head_in_ga(x1, x2)  =  head_in_ga(x1)
[]  =  []
head_out_ga(x1, x2)  =  head_out_ga
.(x1, x2)  =  .(x1, x2)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x2, x5)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x5)
U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x6)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
lessleaves_in_aa(x1, x2)  =  lessleaves_in_aa
lessleaves_out_aa(x1, x2)  =  lessleaves_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x2, x5)

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

(127) UsableRulesProof (EQUIVALENT transformation)

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

(128) Obligation:

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

APP_IN_GGA(X, Y, cons(U, Z)) → U5_GGA(X, Y, U, Z, head_in_ga(X, U))
U5_GGA(X, Y, U, Z, head_out_ga(X, U)) → U6_GGA(X, Y, U, Z, tail_in_ga(X, V))
U6_GGA(X, Y, U, Z, tail_out_ga(X, V)) → APP_IN_GGA(V, Y, Z)

The TRS R consists of the following rules:

head_in_ga([], X3) → head_out_ga([], X3)
head_in_ga(.(X, X4), X) → head_out_ga(.(X, X4), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X5, X), X) → tail_out_ga(.(X5, X), X)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
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)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x2, x5)

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

(129) PiDPToQDPProof (SOUND transformation)

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

(130) Obligation:

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

APP_IN_GGA(X, Y) → U5_GGA(X, Y, head_in_ga(X))
U5_GGA(X, Y, head_out_ga) → U6_GGA(Y, tail_in_ga(X))
U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(X, X4)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(X5, X)) → tail_out_ga(X)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(131) 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:

tail_in_ga(.(X5, X)) → tail_out_ga(X)
head_in_ga(.(X, X4)) → head_out_ga

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + 2·x₂   
POL(APP_IN_GGA(x₁, x₂)) = 2·x₁ + x₂   
POL(U5_GGA(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U6_GGA(x₁, x₂)) = x₁ + x₂   
POL([]) = 0   
POL(head_in_ga(x₁)) = x₁   
POL(head_out_ga) = 0   
POL(tail_in_ga(x₁)) = x₁   
POL(tail_out_ga(x₁)) = 2·x₁   

(132) Obligation:

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

APP_IN_GGA(X, Y) → U5_GGA(X, Y, head_in_ga(X))
U5_GGA(X, Y, head_out_ga) → U6_GGA(Y, tail_in_ga(X))
U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)

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)

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

(133) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule APP_IN_GGA(X, Y) → U5_GGA(X, Y, head_in_ga(X)) at position [2] we obtained the following new rules [LPAR04]:

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)

(134) Obligation:

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

U5_GGA(X, Y, head_out_ga) → U6_GGA(Y, tail_in_ga(X))
U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)

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)

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

(135) 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.

(136) Obligation:

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

U5_GGA(X, Y, head_out_ga) → U6_GGA(Y, tail_in_ga(X))
U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(137) 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].

head_in_ga(x0)

(138) Obligation:

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

U5_GGA(X, Y, head_out_ga) → U6_GGA(Y, tail_in_ga(X))
U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])

The set Q consists of the following terms:

tail_in_ga(x0)

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

(139) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_GGA(X, Y, head_out_ga) → U6_GGA(Y, tail_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

U5_GGA([], y1, head_out_ga) → U6_GGA(y1, tail_out_ga([]))

(140) Obligation:

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

U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)
U5_GGA([], y1, head_out_ga) → U6_GGA(y1, tail_out_ga([]))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])

The set Q consists of the following terms:

tail_in_ga(x0)

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

(141) 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.

(142) Obligation:

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

U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)
U5_GGA([], y1, head_out_ga) → U6_GGA(y1, tail_out_ga([]))

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

tail_in_ga(x0)

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

(143) 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].

tail_in_ga(x0)

(144) Obligation:

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

U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y)
APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)
U5_GGA([], y1, head_out_ga) → U6_GGA(y1, tail_out_ga([]))

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

(145) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GGA(Y, tail_out_ga(V)) → APP_IN_GGA(V, Y) we obtained the following new rules [LPAR04]:

U6_GGA(z0, tail_out_ga([])) → APP_IN_GGA([], z0)

(146) Obligation:

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

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)
U5_GGA([], y1, head_out_ga) → U6_GGA(y1, tail_out_ga([]))
U6_GGA(z0, tail_out_ga([])) → APP_IN_GGA([], z0)

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

(147) 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 = U5_GGA([], y1, head_out_ga) evaluates to t =U5_GGA([], y1, head_out_ga)

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

U5_GGA([], y1, head_out_ga) → U6_GGA(y1, tail_out_ga([]))
with rule U5_GGA([], y1', head_out_ga) → U6_GGA(y1', tail_out_ga([])) at position [] and matcher [y1' / y1]

U6_GGA(y1, tail_out_ga([])) → APP_IN_GGA([], y1)
with rule U6_GGA(z0, tail_out_ga([])) → APP_IN_GGA([], z0) at position [] and matcher [z0 / y1]

APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)
with rule APP_IN_GGA([], y1) → U5_GGA([], y1, head_out_ga)

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.

(148) FALSE