Left Termination proof of /tmp/tmpsV74NQ/blist.pl

Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 CutEliminatorProof (⇐)

↳2 Prolog

↳3 UndefinedPredicateHandlerProof (⇐)

↳4 Prolog

  ↳5 PrologToPiTRSProof (⇐)

    ↳6 PiTRS

    ↳7 DependencyPairsProof (⇔)

    ↳8 PiDP

    ↳9 DependencyGraphProof (⇔)

    ↳10 AND

      ↳11 PiDP

        ↳12 UsableRulesProof (⇔)

        ↳13 PiDP

        ↳14 PiDPToQDPProof (⇐)

        ↳15 QDP

        ↳16 UsableRulesReductionPairsProof (⇔)

        ↳17 QDP

        ↳18 Narrowing (⇐)

        ↳19 QDP

        ↳20 UsableRulesProof (⇔)

        ↳21 QDP

        ↳22 QReductionProof (⇔)

        ↳23 QDP

        ↳24 Instantiation (⇔)

        ↳25 QDP

        ↳26 NonTerminationProof (⇔)

        ↳27 FALSE

      ↳28 PiDP

        ↳29 UsableRulesProof (⇔)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇐)

        ↳32 QDP

        ↳33 UsableRulesReductionPairsProof (⇔)

        ↳34 QDP

        ↳35 Narrowing (⇐)

        ↳36 QDP

        ↳37 UsableRulesProof (⇔)

        ↳38 QDP

        ↳39 QReductionProof (⇔)

        ↳40 QDP

        ↳41 Instantiation (⇔)

        ↳42 QDP

        ↳43 NonTerminationProof (⇔)

        ↳44 FALSE

  ↳45 PrologToPiTRSProof (⇐)

    ↳46 PiTRS

    ↳47 DependencyPairsProof (⇔)

    ↳48 PiDP

    ↳49 DependencyGraphProof (⇔)

    ↳50 AND

      ↳51 PiDP

        ↳52 UsableRulesProof (⇔)

        ↳53 PiDP

        ↳54 PiDPToQDPProof (⇐)

        ↳55 QDP

        ↳56 Narrowing (⇐)

        ↳57 QDP

        ↳58 UsableRulesProof (⇔)

        ↳59 QDP

        ↳60 QReductionProof (⇔)

        ↳61 QDP

        ↳62 Instantiation (⇔)

        ↳63 QDP

        ↳64 DependencyGraphProof (⇔)

        ↳65 AND

          ↳66 QDP

            ↳67 NonTerminationProof (⇔)

            ↳68 FALSE

          ↳69 QDP

            ↳70 QDPSizeChangeProof (⇔)

            ↳71 TRUE

      ↳72 PiDP

        ↳73 UsableRulesProof (⇔)

        ↳74 PiDP

        ↳75 PiDPToQDPProof (⇐)

        ↳76 QDP

        ↳77 Narrowing (⇐)

        ↳78 QDP

        ↳79 UsableRulesProof (⇔)

        ↳80 QDP

        ↳81 QReductionProof (⇔)

        ↳82 QDP

        ↳83 Instantiation (⇔)

        ↳84 QDP

        ↳85 DependencyGraphProof (⇔)

        ↳86 AND

          ↳87 QDP

            ↳88 NonTerminationProof (⇔)

            ↳89 FALSE

          ↳90 QDP

            ↳91 QDPSizeChangeProof (⇔)

            ↳92 TRUE

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), list(Xs)).
list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), list(Xs)).
list([]) :- true.
list(X) :- ','(tail(X, T), list(T)).
s2l(0, L) :- eq(L, []).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

(5) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2l_in: (b,f)
list_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

(7) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U6_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U7_GA(X, X1, Xs, p_out_ga(X, P)) → U8_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, list_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → LIST_IN_G(Xs)
LIST_IN_G([]) → U3_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U4_G(X, tail_out_ga(X, T)) → U5_G(X, list_in_g(T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
0 = 0
U6_ga(x1, x2) = U6_ga(x2)
eq_in_ag(x1, x2) = eq_in_ag(x2)
eq_out_ag(x1, x2) = eq_out_ag(x1)
[] = []
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
U7_ga(x1, x2, x3, x4) = U7_ga(x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U8_ga(x1, x2, x3, x4) = U8_ga(x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x2)
list_in_g(x1) = list_in_g(x1)
U3_g(x1) = U3_g(x1)
true_in_ = true_in_
true_out_ = true_out_
list_out_g(x1) = list_out_g
U4_g(x1, x2) = U4_g(x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
U5_g(x1, x2) = U5_g(x2)
goal_out_g(x1) = goal_out_g
GOAL_IN_G(x1) = GOAL_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U6_GA(x1, x2) = U6_GA(x2)
EQ_IN_AG(x1, x2) = EQ_IN_AG(x2)
U7_GA(x1, x2, x3, x4) = U7_GA(x4)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U8_GA(x1, x2, x3, x4) = U8_GA(x4)
U2_G(x1, x2) = U2_G(x2)
LIST_IN_G(x1) = LIST_IN_G(x1)
U3_G(x1) = U3_G(x1)
TRUE_IN_ = TRUE_IN_
U4_G(x1, x2) = U4_G(x2)
TAIL_IN_GA(x1, x2) = TAIL_IN_GA(x1)
U5_G(x1, x2) = U5_G(x2)

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

(8) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U6_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U7_GA(X, X1, Xs, p_out_ga(X, P)) → U8_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, list_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → LIST_IN_G(Xs)
LIST_IN_G([]) → U3_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U4_G(X, tail_out_ga(X, T)) → U5_G(X, list_in_g(T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
LIST_IN_G(x1) = LIST_IN_G(x1)
U4_G(x1, x2) = U4_G(x2)

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

LIST_IN_G(X) → U4_G(tail_in_ga(X))
U4_G(tail_out_ga(T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(16) 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(.(Xs)) → tail_out_ga(Xs)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁)) = 2·x₁
POL(LIST_IN_G(x₁)) = x₁
POL(U4_G(x₁)) = x₁
POL([]) = 0
POL(tail_in_ga(x₁)) = x₁
POL(tail_out_ga(x₁)) = 2·x₁

(17) Obligation:

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

LIST_IN_G(X) → U4_G(tail_in_ga(X))
U4_G(tail_out_ga(T)) → LIST_IN_G(T)

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.

(18) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LIST_IN_G(X) → U4_G(tail_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

LIST_IN_G([]) → U4_G(tail_out_ga([]))

(19) Obligation:

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

U4_G(tail_out_ga(T)) → LIST_IN_G(T)
LIST_IN_G([]) → U4_G(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.

(20) 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.

(21) Obligation:

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

U4_G(tail_out_ga(T)) → LIST_IN_G(T)
LIST_IN_G([]) → U4_G(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.

(22) 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)

(23) Obligation:

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

U4_G(tail_out_ga(T)) → LIST_IN_G(T)
LIST_IN_G([]) → U4_G(tail_out_ga([]))

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

(24) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_G(tail_out_ga(T)) → LIST_IN_G(T) we obtained the following new rules [LPAR04]:

U4_G(tail_out_ga([])) → LIST_IN_G([])

(25) Obligation:

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

LIST_IN_G([]) → U4_G(tail_out_ga([]))
U4_G(tail_out_ga([])) → LIST_IN_G([])

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 = U4_G(tail_out_ga([])) evaluates to t =U4_G(tail_out_ga([]))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U4_G(tail_out_ga([])) → LIST_IN_G([])
with rule U4_G(tail_out_ga([])) → LIST_IN_G([]) at position [] and matcher [ ]

LIST_IN_G([]) → U4_G(tail_out_ga([]))
with rule LIST_IN_G([]) → U4_G(tail_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.

(27) FALSE

(28) Obligation:

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

S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
0 = 0
U6_ga(x1, x2) = U6_ga(x2)
eq_in_ag(x1, x2) = eq_in_ag(x2)
eq_out_ag(x1, x2) = eq_out_ag(x1)
[] = []
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
U7_ga(x1, x2, x3, x4) = U7_ga(x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U8_ga(x1, x2, x3, x4) = U8_ga(x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x2)
list_in_g(x1) = list_in_g(x1)
U3_g(x1) = U3_g(x1)
true_in_ = true_in_
true_out_ = true_out_
list_out_g(x1) = list_out_g
U4_g(x1, x2) = U4_g(x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
U5_g(x1, x2) = U5_g(x2)
goal_out_g(x1) = goal_out_g
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U7_GA(x1, x2, x3, x4) = U7_GA(x4)

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:

S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
.(x1, x2) = .(x2)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U7_GA(x1, x2, x3, x4) = U7_GA(x4)

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:

S2L_IN_GA(X) → U7_GA(p_in_ga(X))
U7_GA(p_out_ga(P)) → S2L_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(33) UsableRulesReductionPairsProof (EQUIVALENT transformation)

p_in_ga(s(X)) → p_out_ga(X)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0
POL(S2L_IN_GA(x₁)) = x₁
POL(U7_GA(x₁)) = x₁
POL(p_in_ga(x₁)) = x₁
POL(p_out_ga(x₁)) = 2·x₁
POL(s(x₁)) = 2·x₁

(34) Obligation:

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

S2L_IN_GA(X) → U7_GA(p_in_ga(X))
U7_GA(p_out_ga(P)) → S2L_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(35) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2L_IN_GA(X) → U7_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2L_IN_GA(0) → U7_GA(p_out_ga(0))

(36) Obligation:

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

U7_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U7_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

U7_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U7_GA(p_out_ga(0))

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

p_in_ga(x0)

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

(39) QReductionProof (EQUIVALENT transformation)

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

p_in_ga(x0)

(40) Obligation:

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

U7_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U7_GA(p_out_ga(0))

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

(41) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_GA(p_out_ga(P)) → S2L_IN_GA(P) we obtained the following new rules [LPAR04]:

U7_GA(p_out_ga(0)) → S2L_IN_GA(0)

(42) Obligation:

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

S2L_IN_GA(0) → U7_GA(p_out_ga(0))
U7_GA(p_out_ga(0)) → S2L_IN_GA(0)

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

(43) 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 = U7_GA(p_out_ga(0)) evaluates to t =U7_GA(p_out_ga(0))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U7_GA(p_out_ga(0)) → S2L_IN_GA(0)
with rule U7_GA(p_out_ga(0)) → S2L_IN_GA(0) at position [] and matcher [ ]

S2L_IN_GA(0) → U7_GA(p_out_ga(0))
with rule S2L_IN_GA(0) → U7_GA(p_out_ga(0))

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.

(44) FALSE

(45) PrologToPiTRSProof (SOUND transformation)

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(46) Obligation:

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

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

(47) 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:

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U6_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U7_GA(X, X1, Xs, p_out_ga(X, P)) → U8_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, list_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → LIST_IN_G(Xs)
LIST_IN_G([]) → U3_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U4_G(X, tail_out_ga(X, T)) → U5_G(X, list_in_g(T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
0 = 0
U6_ga(x1, x2) = U6_ga(x2)
eq_in_ag(x1, x2) = eq_in_ag(x2)
eq_out_ag(x1, x2) = eq_out_ag(x1, x2)
[] = []
s2l_out_ga(x1, x2) = s2l_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
U8_ga(x1, x2, x3, x4) = U8_ga(x1, x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x1, x2)
list_in_g(x1) = list_in_g(x1)
U3_g(x1) = U3_g(x1)
true_in_ = true_in_
true_out_ = true_out_
list_out_g(x1) = list_out_g(x1)
U4_g(x1, x2) = U4_g(x1, x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
U5_g(x1, x2) = U5_g(x1, x2)
goal_out_g(x1) = goal_out_g(x1)
GOAL_IN_G(x1) = GOAL_IN_G(x1)
U1_G(x1, x2) = U1_G(x1, x2)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U6_GA(x1, x2) = U6_GA(x2)
EQ_IN_AG(x1, x2) = EQ_IN_AG(x2)
U7_GA(x1, x2, x3, x4) = U7_GA(x1, x4)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U8_GA(x1, x2, x3, x4) = U8_GA(x1, x4)
U2_G(x1, x2) = U2_G(x1, x2)
LIST_IN_G(x1) = LIST_IN_G(x1)
U3_G(x1) = U3_G(x1)
TRUE_IN_ = TRUE_IN_
U4_G(x1, x2) = U4_G(x1, x2)
TAIL_IN_GA(x1, x2) = TAIL_IN_GA(x1)
U5_G(x1, x2) = U5_G(x1, x2)

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

(48) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U6_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U7_GA(X, X1, Xs, p_out_ga(X, P)) → U8_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, list_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → LIST_IN_G(Xs)
LIST_IN_G([]) → U3_G(true_in_)
LIST_IN_G([]) → TRUE_IN_
LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
LIST_IN_G(X) → TAIL_IN_GA(X, T)
U4_G(X, tail_out_ga(X, T)) → U5_G(X, list_in_g(T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

(49) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(50) Complex Obligation (AND)

(51) Obligation:

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

LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

(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:

LIST_IN_G(X) → U4_G(X, tail_in_ga(X, T))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
LIST_IN_G(x1) = LIST_IN_G(x1)
U4_G(x1, x2) = U4_G(x1, x2)

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:

LIST_IN_G(X) → U4_G(X, tail_in_ga(X))
U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(56) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LIST_IN_G(X) → U4_G(X, tail_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

LIST_IN_G([]) → U4_G([], tail_out_ga([], []))
LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))

(57) Obligation:

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

U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U4_G([], tail_out_ga([], []))
LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

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:

U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U4_G([], tail_out_ga([], []))
LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))

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

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].

tail_in_ga(x0)

(61) Obligation:

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

U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T)
LIST_IN_G([]) → U4_G([], tail_out_ga([], []))
LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))

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

(62) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_G(X, tail_out_ga(X, T)) → LIST_IN_G(T) we obtained the following new rules [LPAR04]:

U4_G([], tail_out_ga([], [])) → LIST_IN_G([])
U4_G(.(z0), tail_out_ga(.(z0), z0)) → LIST_IN_G(z0)

(63) Obligation:

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

LIST_IN_G([]) → U4_G([], tail_out_ga([], []))
LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))
U4_G([], tail_out_ga([], [])) → LIST_IN_G([])
U4_G(.(z0), tail_out_ga(.(z0), z0)) → LIST_IN_G(z0)

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

(64) DependencyGraphProof (EQUIVALENT transformation)

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

(65) Complex Obligation (AND)

(66) Obligation:

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

U4_G([], tail_out_ga([], [])) → LIST_IN_G([])
LIST_IN_G([]) → U4_G([], tail_out_ga([], []))

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

(67) 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 = LIST_IN_G([]) evaluates to t =LIST_IN_G([])

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

LIST_IN_G([]) → U4_G([], tail_out_ga([], []))
with rule LIST_IN_G([]) → U4_G([], tail_out_ga([], [])) at position [] and matcher [ ]

U4_G([], tail_out_ga([], [])) → LIST_IN_G([])
with rule U4_G([], tail_out_ga([], [])) → LIST_IN_G([])

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.

(68) FALSE

(69) Obligation:

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

LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))
U4_G(.(z0), tail_out_ga(.(z0), z0)) → LIST_IN_G(z0)

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

(70) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

U4_G(.(z0), tail_out_ga(.(z0), z0)) → LIST_IN_G(z0)
The graph contains the following edges 1 > 1, 2 > 1
LIST_IN_G(.(x0)) → U4_G(.(x0), tail_out_ga(.(x0), x0))
The graph contains the following edges 1 >= 1

(71) TRUE

(72) Obligation:

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

S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U6_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U6_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U7_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U7_ga(X, X1, Xs, p_out_ga(X, P)) → U8_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U8_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, list_in_g(Xs))
list_in_g([]) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → list_out_g([])
list_in_g(X) → U4_g(X, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X2, Xs), Xs) → tail_out_ga(.(X2, Xs), Xs)
U4_g(X, tail_out_ga(X, T)) → U5_g(X, list_in_g(T))
U5_g(X, list_out_g(T)) → list_out_g(X)
U2_g(X, list_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
0 = 0
U6_ga(x1, x2) = U6_ga(x2)
eq_in_ag(x1, x2) = eq_in_ag(x2)
eq_out_ag(x1, x2) = eq_out_ag(x1, x2)
[] = []
s2l_out_ga(x1, x2) = s2l_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
U8_ga(x1, x2, x3, x4) = U8_ga(x1, x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x1, x2)
list_in_g(x1) = list_in_g(x1)
U3_g(x1) = U3_g(x1)
true_in_ = true_in_
true_out_ = true_out_
list_out_g(x1) = list_out_g(x1)
U4_g(x1, x2) = U4_g(x1, x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
U5_g(x1, x2) = U5_g(x1, x2)
goal_out_g(x1) = goal_out_g(x1)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U7_GA(x1, x2, x3, x4) = U7_GA(x1, x4)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

S2L_IN_GA(X, .(X1, Xs)) → U7_GA(X, X1, Xs, p_in_ga(X, P))
U7_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
.(x1, x2) = .(x2)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U7_GA(x1, x2, x3, x4) = U7_GA(x1, x4)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

S2L_IN_GA(X) → U7_GA(X, p_in_ga(X))
U7_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(77) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2L_IN_GA(X) → U7_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))

(78) Obligation:

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

U7_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(79) 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.

(80) Obligation:

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

U7_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))

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

p_in_ga(x0)

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

(81) 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].

p_in_ga(x0)

(82) Obligation:

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

U7_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))

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

(83) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P) we obtained the following new rules [LPAR04]:

U7_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
U7_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)

(84) Obligation:

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

S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))
U7_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
U7_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)

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

(85) DependencyGraphProof (EQUIVALENT transformation)

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

(86) Complex Obligation (AND)

(87) Obligation:

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

U7_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))

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

(88) 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 = S2L_IN_GA(0) evaluates to t =S2L_IN_GA(0)

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0))
with rule S2L_IN_GA(0) → U7_GA(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U7_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
with rule U7_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)

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.

(89) FALSE

(90) Obligation:

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

S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))
U7_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)

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

(91) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

U7_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)
The graph contains the following edges 1 > 1, 2 > 1
S2L_IN_GA(s(x0)) → U7_GA(s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1