Left Termination proof of /tmp/tmpmUmvXf/bappend.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 PrologToPiTRSProof (⇐)

    ↳4 PiTRS

    ↳5 DependencyPairsProof (⇔)

    ↳6 PiDP

    ↳7 DependencyGraphProof (⇔)

    ↳8 AND

      ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 UsableRulesReductionPairsProof (⇔)

        ↳15 QDP

        ↳16 Narrowing (⇐)

        ↳17 QDP

        ↳18 UsableRulesProof (⇔)

        ↳19 QDP

        ↳20 QReductionProof (⇔)

        ↳21 QDP

        ↳22 Narrowing (⇐)

        ↳23 QDP

        ↳24 UsableRulesProof (⇔)

        ↳25 QDP

        ↳26 QReductionProof (⇔)

        ↳27 QDP

        ↳28 Instantiation (⇔)

        ↳29 QDP

        ↳30 NonTerminationProof (⇔)

        ↳31 FALSE

      ↳32 PiDP

        ↳33 UsableRulesProof (⇔)

        ↳34 PiDP

        ↳35 PiDPToQDPProof (⇐)

        ↳36 QDP

        ↳37 UsableRulesReductionPairsProof (⇔)

        ↳38 QDP

        ↳39 Narrowing (⇐)

        ↳40 QDP

        ↳41 UsableRulesProof (⇔)

        ↳42 QDP

        ↳43 QReductionProof (⇔)

        ↳44 QDP

        ↳45 Instantiation (⇔)

        ↳46 QDP

        ↳47 NonTerminationProof (⇔)

        ↳48 FALSE

  ↳49 PrologToPiTRSProof (⇐)

    ↳50 PiTRS

    ↳51 DependencyPairsProof (⇔)

    ↳52 PiDP

    ↳53 DependencyGraphProof (⇔)

    ↳54 AND

      ↳55 PiDP

        ↳56 UsableRulesProof (⇔)

        ↳57 PiDP

        ↳58 PiDPToQDPProof (⇐)

        ↳59 QDP

        ↳60 Narrowing (⇐)

        ↳61 QDP

        ↳62 UsableRulesProof (⇔)

        ↳63 QDP

        ↳64 QReductionProof (⇔)

        ↳65 QDP

        ↳66 Narrowing (⇐)

        ↳67 QDP

        ↳68 UsableRulesProof (⇔)

        ↳69 QDP

        ↳70 QReductionProof (⇔)

        ↳71 QDP

        ↳72 Instantiation (⇔)

        ↳73 QDP

        ↳74 DependencyGraphProof (⇔)

        ↳75 AND

          ↳76 QDP

            ↳77 NonTerminationProof (⇔)

            ↳78 FALSE

          ↳79 QDP

            ↳80 QDPSizeChangeProof (⇔)

            ↳81 TRUE

      ↳82 PiDP

        ↳83 UsableRulesProof (⇔)

        ↳84 PiDP

        ↳85 PiDPToQDPProof (⇐)

        ↳86 QDP

        ↳87 Narrowing (⇐)

        ↳88 QDP

        ↳89 UsableRulesProof (⇔)

        ↳90 QDP

        ↳91 QReductionProof (⇔)

        ↳92 QDP

        ↳93 Instantiation (⇔)

        ↳94 QDP

        ↳95 DependencyGraphProof (⇔)

        ↳96 AND

          ↳97 QDP

            ↳98 NonTerminationProof (⇔)

            ↳99 FALSE

          ↳100 QDP

            ↳101 QDPSizeChangeProof (⇔)

            ↳102 TRUE

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), append(Xs, X1, X2)).
append([], Y, Z) :- ','(!, eq(Y, Z)).
append(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), append(T, Y, Z))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, 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), append(Xs, X1, X2)).
append([], Y, Z) :- eq(Y, Z).
append(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), append(T, Y, Z))).
s2l(0, L) :- eq(L, []).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(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:
goal_in: (b)
s2l_in: (b,f)
append_in: (b,f,f)
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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

(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:

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) → U7_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U8_GA(X, X3, Xs, p_out_ga(X, P)) → U9_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U8_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, append_in_gaa(Xs, X1, X2))
U1_G(X, s2l_out_ga(X, Xs)) → APPEND_IN_GAA(Xs, X1, X2)
APPEND_IN_GAA([], Y, Z) → U3_GAA(Y, Z, eq_in_aa(Y, Z))
APPEND_IN_GAA([], Y, Z) → EQ_IN_AA(Y, Z)
APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
APPEND_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U6_GAA(X, Y, H, Z, append_in_gaa(T, Y, Z))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → 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
U7_ga(x1, x2) = U7_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)
U8_ga(x1, x2, x3, x4) = U8_ga(x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U9_ga(x1, x2, x3, x4) = U9_ga(x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
U3_gaa(x1, x2, x3) = U3_gaa(x3)
eq_in_aa(x1, x2) = eq_in_aa
eq_out_aa(x1, x2) = eq_out_aa
append_out_gaa(x1, x2, x3) = append_out_gaa
U4_gaa(x1, x2, x3, x4, x5) = U4_gaa(x1, x5)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
U6_gaa(x1, x2, x3, x4, x5) = U6_gaa(x5)
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)
U7_GA(x1, x2) = U7_GA(x2)
EQ_IN_AG(x1, x2) = EQ_IN_AG(x2)
U8_GA(x1, x2, x3, x4) = U8_GA(x4)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U9_GA(x1, x2, x3, x4) = U9_GA(x4)
U2_G(x1, x2) = U2_G(x2)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U3_GAA(x1, x2, x3) = U3_GAA(x3)
EQ_IN_AA(x1, x2) = EQ_IN_AA
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
HEAD_IN_GA(x1, x2) = HEAD_IN_GA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x5)
TAIL_IN_GA(x1, x2) = TAIL_IN_GA(x1)
U6_GAA(x1, x2, x3, x4, x5) = U6_GAA(x5)

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

(6) 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) → U7_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U8_GA(X, X3, Xs, p_out_ga(X, P)) → U9_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U8_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, append_in_gaa(Xs, X1, X2))
U1_G(X, s2l_out_ga(X, Xs)) → APPEND_IN_GAA(Xs, X1, X2)
APPEND_IN_GAA([], Y, Z) → U3_GAA(Y, Z, eq_in_aa(Y, Z))
APPEND_IN_GAA([], Y, Z) → EQ_IN_AA(Y, Z)
APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
APPEND_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U6_GAA(X, Y, H, Z, append_in_gaa(T, Y, Z))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 13 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → 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
U7_ga(x1, x2) = U7_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)
U8_ga(x1, x2, x3, x4) = U8_ga(x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U9_ga(x1, x2, x3, x4) = U9_ga(x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
U3_gaa(x1, x2, x3) = U3_gaa(x3)
eq_in_aa(x1, x2) = eq_in_aa
eq_out_aa(x1, x2) = eq_out_aa
append_out_gaa(x1, x2, x3) = append_out_gaa
U4_gaa(x1, x2, x3, x4, x5) = U4_gaa(x1, x5)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
U6_gaa(x1, x2, x3, x4, x5) = U6_gaa(x5)
goal_out_g(x1) = goal_out_g
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(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:

APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(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:

APPEND_IN_GAA(X) → U4_GAA(X, head_in_ga(X))
U4_GAA(X, head_out_ga) → U5_GAA(tail_in_ga(X))
U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(X5)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

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.

(14) 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)
head_in_ga(.(X5)) → head_out_ga

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁)) = 2·x₁
POL(APPEND_IN_GAA(x₁)) = 2·x₁
POL(U4_GAA(x₁, x₂)) = x₁ + x₂
POL(U5_GAA(x₁)) = x₁
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

(15) Obligation:

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

APPEND_IN_GAA(X) → U4_GAA(X, head_in_ga(X))
U4_GAA(X, head_out_ga) → U5_GAA(tail_in_ga(X))
U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(16) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule APPEND_IN_GAA(X) → U4_GAA(X, head_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

APPEND_IN_GAA([]) → U4_GAA([], head_out_ga)

(17) Obligation:

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

U4_GAA(X, head_out_ga) → U5_GAA(tail_in_ga(X))
U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], 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.

(18) 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.

(19) Obligation:

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

U4_GAA(X, head_out_ga) → U5_GAA(tail_in_ga(X))
U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], 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.

(20) 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)

(21) Obligation:

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

U4_GAA(X, head_out_ga) → U5_GAA(tail_in_ga(X))
U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], 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.

(22) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U4_GAA(X, head_out_ga) → U5_GAA(tail_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

U4_GAA([], head_out_ga) → U5_GAA(tail_out_ga([]))

(23) Obligation:

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

U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga)
U4_GAA([], head_out_ga) → U5_GAA(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.

(24) 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.

(25) Obligation:

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

U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga)
U4_GAA([], head_out_ga) → U5_GAA(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.

(26) 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)

(27) Obligation:

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

U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga)
U4_GAA([], head_out_ga) → U5_GAA(tail_out_ga([]))

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

(28) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GAA(tail_out_ga(T)) → APPEND_IN_GAA(T) we obtained the following new rules [LPAR04]:

U5_GAA(tail_out_ga([])) → APPEND_IN_GAA([])

(29) Obligation:

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

APPEND_IN_GAA([]) → U4_GAA([], head_out_ga)
U4_GAA([], head_out_ga) → U5_GAA(tail_out_ga([]))
U5_GAA(tail_out_ga([])) → APPEND_IN_GAA([])

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

(30) 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_GAA([], head_out_ga) evaluates to t =U4_GAA([], head_out_ga)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U4_GAA([], head_out_ga) → U5_GAA(tail_out_ga([]))
with rule U4_GAA([], head_out_ga) → U5_GAA(tail_out_ga([])) at position [] and matcher [ ]

U5_GAA(tail_out_ga([])) → APPEND_IN_GAA([])
with rule U5_GAA(tail_out_ga([])) → APPEND_IN_GAA([]) at position [] and matcher [ ]

APPEND_IN_GAA([]) → U4_GAA([], head_out_ga)
with rule APPEND_IN_GAA([]) → U4_GAA([], 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.

(31) FALSE

(32) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
U8_GA(X, X3, 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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
U8_GA(X, X3, 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)
U8_GA(x1, x2, x3, x4) = U8_GA(x4)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

S2L_IN_GA(X) → U8_GA(p_in_ga(X))
U8_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.

(37) 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(U8_GA(x₁)) = x₁
POL(p_in_ga(x₁)) = x₁
POL(p_out_ga(x₁)) = 2·x₁
POL(s(x₁)) = 2·x₁

(38) Obligation:

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

S2L_IN_GA(X) → U8_GA(p_in_ga(X))
U8_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.

(39) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2L_IN_GA(X) → U8_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2L_IN_GA(0) → U8_GA(p_out_ga(0))

(40) Obligation:

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

U8_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U8_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.

(41) 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.

(42) Obligation:

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

U8_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U8_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.

(43) 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)

(44) Obligation:

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

U8_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U8_GA(p_out_ga(0))

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

(45) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U8_GA(p_out_ga(P)) → S2L_IN_GA(P) we obtained the following new rules [LPAR04]:

U8_GA(p_out_ga(0)) → S2L_IN_GA(0)

(46) Obligation:

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

S2L_IN_GA(0) → U8_GA(p_out_ga(0))
U8_GA(p_out_ga(0)) → S2L_IN_GA(0)

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

(47) 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 = U8_GA(p_out_ga(0)) evaluates to t =U8_GA(p_out_ga(0))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U8_GA(p_out_ga(0)) → S2L_IN_GA(0)
with rule U8_GA(p_out_ga(0)) → S2L_IN_GA(0) at position [] and matcher [ ]

S2L_IN_GA(0) → U8_GA(p_out_ga(0))
with rule S2L_IN_GA(0) → U8_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.

(48) FALSE

(49) PrologToPiTRSProof (SOUND transformation)

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(50) 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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

(51) 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) → U7_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U8_GA(X, X3, Xs, p_out_ga(X, P)) → U9_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U8_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, append_in_gaa(Xs, X1, X2))
U1_G(X, s2l_out_ga(X, Xs)) → APPEND_IN_GAA(Xs, X1, X2)
APPEND_IN_GAA([], Y, Z) → U3_GAA(Y, Z, eq_in_aa(Y, Z))
APPEND_IN_GAA([], Y, Z) → EQ_IN_AA(Y, Z)
APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
APPEND_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U6_GAA(X, Y, H, Z, append_in_gaa(T, Y, Z))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → 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
U7_ga(x1, x2) = U7_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)
U8_ga(x1, x2, x3, x4) = U8_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)
U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x1, x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
U3_gaa(x1, x2, x3) = U3_gaa(x3)
eq_in_aa(x1, x2) = eq_in_aa
eq_out_aa(x1, x2) = eq_out_aa
append_out_gaa(x1, x2, x3) = append_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5) = U4_gaa(x1, x5)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga(x1)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
U6_gaa(x1, x2, x3, x4, x5) = U6_gaa(x1, x5)
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)
U7_GA(x1, x2) = U7_GA(x2)
EQ_IN_AG(x1, x2) = EQ_IN_AG(x2)
U8_GA(x1, x2, x3, x4) = U8_GA(x1, x4)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4)
U2_G(x1, x2) = U2_G(x1, x2)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U3_GAA(x1, x2, x3) = U3_GAA(x3)
EQ_IN_AA(x1, x2) = EQ_IN_AA
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
HEAD_IN_GA(x1, x2) = HEAD_IN_GA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)
TAIL_IN_GA(x1, x2) = TAIL_IN_GA(x1)
U6_GAA(x1, x2, x3, x4, x5) = U6_GAA(x1, x5)

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

(52) 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) → U7_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U8_GA(X, X3, Xs, p_out_ga(X, P)) → U9_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U8_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, append_in_gaa(Xs, X1, X2))
U1_G(X, s2l_out_ga(X, Xs)) → APPEND_IN_GAA(Xs, X1, X2)
APPEND_IN_GAA([], Y, Z) → U3_GAA(Y, Z, eq_in_aa(Y, Z))
APPEND_IN_GAA([], Y, Z) → EQ_IN_AA(Y, Z)
APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
APPEND_IN_GAA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → U6_GAA(X, Y, H, Z, append_in_gaa(T, Y, Z))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

(53) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 13 less nodes.

(54) Complex Obligation (AND)

(55) Obligation:

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

APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → 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
U7_ga(x1, x2) = U7_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)
U8_ga(x1, x2, x3, x4) = U8_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)
U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4)
.(x1, x2) = .(x2)
U2_g(x1, x2) = U2_g(x1, x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
U3_gaa(x1, x2, x3) = U3_gaa(x3)
eq_in_aa(x1, x2) = eq_in_aa
eq_out_aa(x1, x2) = eq_out_aa
append_out_gaa(x1, x2, x3) = append_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5) = U4_gaa(x1, x5)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga(x1)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
U6_gaa(x1, x2, x3, x4, x5) = U6_gaa(x1, x5)
goal_out_g(x1) = goal_out_g(x1)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)

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

(56) UsableRulesProof (EQUIVALENT transformation)

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

(57) Obligation:

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

APPEND_IN_GAA(X, Y, .(H, Z)) → U4_GAA(X, Y, H, Z, head_in_ga(X, H))
U4_GAA(X, Y, H, Z, head_out_ga(X, H)) → U5_GAA(X, Y, H, Z, tail_in_ga(X, T))
U5_GAA(X, Y, H, Z, tail_out_ga(X, T)) → APPEND_IN_GAA(T, Y, Z)

The TRS R consists of the following rules:

head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga(x1)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)

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

(58) PiDPToQDPProof (SOUND transformation)

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

(59) Obligation:

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

APPEND_IN_GAA(X) → U4_GAA(X, head_in_ga(X))
U4_GAA(X, head_out_ga(X)) → U5_GAA(X, tail_in_ga(X))
U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)

The TRS R consists of the following rules:

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

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) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule APPEND_IN_GAA(X) → U4_GAA(X, head_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))

(61) Obligation:

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

U4_GAA(X, head_out_ga(X)) → U5_GAA(X, tail_in_ga(X))
U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))

The TRS R consists of the following rules:

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

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.

(62) 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.

(63) Obligation:

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

U4_GAA(X, head_out_ga(X)) → U5_GAA(X, tail_in_ga(X))
U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(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:

head_in_ga(x0)
tail_in_ga(x0)

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

(64) 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)

(65) Obligation:

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

U4_GAA(X, head_out_ga(X)) → U5_GAA(X, tail_in_ga(X))
U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(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.

(66) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U4_GAA(X, head_out_ga(X)) → U5_GAA(X, tail_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
U4_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(x0), tail_out_ga(.(x0), x0))

(67) Obligation:

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

U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))
U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
U4_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(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.

(68) 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.

(69) Obligation:

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

U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))
U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
U4_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(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.

(70) 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)

(71) Obligation:

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

U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T)
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))
U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
U4_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(x0), tail_out_ga(.(x0), x0))

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

(72) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GAA(X, tail_out_ga(X, T)) → APPEND_IN_GAA(T) we obtained the following new rules [LPAR04]:

U5_GAA([], tail_out_ga([], [])) → APPEND_IN_GAA([])
U5_GAA(.(z0), tail_out_ga(.(z0), z0)) → APPEND_IN_GAA(z0)

(73) Obligation:

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

APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))
U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
U4_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(x0), tail_out_ga(.(x0), x0))
U5_GAA([], tail_out_ga([], [])) → APPEND_IN_GAA([])
U5_GAA(.(z0), tail_out_ga(.(z0), z0)) → APPEND_IN_GAA(z0)

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

(74) DependencyGraphProof (EQUIVALENT transformation)

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

(75) Complex Obligation (AND)

(76) Obligation:

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

U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
U5_GAA([], tail_out_ga([], [])) → APPEND_IN_GAA([])
APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))

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

(77) 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_GAA([], tail_out_ga([], [])) evaluates to t =U5_GAA([], tail_out_ga([], []))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U5_GAA([], tail_out_ga([], [])) → APPEND_IN_GAA([])
with rule U5_GAA([], tail_out_ga([], [])) → APPEND_IN_GAA([]) at position [] and matcher [ ]

APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([]))
with rule APPEND_IN_GAA([]) → U4_GAA([], head_out_ga([])) at position [] and matcher [ ]

U4_GAA([], head_out_ga([])) → U5_GAA([], tail_out_ga([], []))
with rule U4_GAA([], head_out_ga([])) → U5_GAA([], 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.

(78) FALSE

(79) Obligation:

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

APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))
U4_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(x0), tail_out_ga(.(x0), x0))
U5_GAA(.(z0), tail_out_ga(.(z0), z0)) → APPEND_IN_GAA(z0)

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

(80) 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_GAA(.(x0), head_out_ga(.(x0))) → U5_GAA(.(x0), tail_out_ga(.(x0), x0))
The graph contains the following edges 1 >= 1, 2 > 1
U5_GAA(.(z0), tail_out_ga(.(z0), z0)) → APPEND_IN_GAA(z0)
The graph contains the following edges 1 > 1, 2 > 1
APPEND_IN_GAA(.(x0)) → U4_GAA(.(x0), head_out_ga(.(x0)))
The graph contains the following edges 1 >= 1

(81) TRUE

(82) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
U8_GA(X, X3, 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) → U7_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U7_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U8_ga(X, X3, 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)
U8_ga(X, X3, Xs, p_out_ga(X, P)) → U9_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U9_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, append_in_gaa(Xs, X1, X2))
append_in_gaa([], Y, Z) → U3_gaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_gaa(Y, Z, eq_out_aa(Y, Z)) → append_out_gaa([], Y, Z)
append_in_gaa(X, Y, .(H, Z)) → U4_gaa(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U4_gaa(X, Y, H, Z, head_out_ga(X, H)) → U5_gaa(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U5_gaa(X, Y, H, Z, tail_out_ga(X, T)) → U6_gaa(X, Y, H, Z, append_in_gaa(T, Y, Z))
U6_gaa(X, Y, H, Z, append_out_gaa(T, Y, Z)) → append_out_gaa(X, Y, .(H, Z))
U2_g(X, append_out_gaa(Xs, X1, X2)) → goal_out_g(X)

(83) UsableRulesProof (EQUIVALENT transformation)

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

(84) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U8_GA(X, X3, Xs, p_in_ga(X, P))
U8_GA(X, X3, 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)
U8_GA(x1, x2, x3, x4) = U8_GA(x1, x4)

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

(85) PiDPToQDPProof (SOUND transformation)

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

(86) Obligation:

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

S2L_IN_GA(X) → U8_GA(X, p_in_ga(X))
U8_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.

(87) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2L_IN_GA(X) → U8_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U8_GA(s(x0), p_out_ga(s(x0), x0))

(88) Obligation:

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

U8_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U8_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.

(89) 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.

(90) Obligation:

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

U8_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U8_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.

(91) 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)

(92) Obligation:

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

U8_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U8_GA(s(x0), p_out_ga(s(x0), x0))

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

(93) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U8_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P) we obtained the following new rules [LPAR04]:

U8_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
U8_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)

(94) Obligation:

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

S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U8_GA(s(x0), p_out_ga(s(x0), x0))
U8_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
U8_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.

(95) DependencyGraphProof (EQUIVALENT transformation)

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

(96) Complex Obligation (AND)

(97) Obligation:

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

U8_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0))

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

(98) 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) → U8_GA(0, p_out_ga(0, 0))
with rule S2L_IN_GA(0) → U8_GA(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U8_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
with rule U8_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.

(99) FALSE

(100) Obligation:

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

S2L_IN_GA(s(x0)) → U8_GA(s(x0), p_out_ga(s(x0), x0))
U8_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.

(101) 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:

U8_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)) → U8_GA(s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1