Left Termination of the query pattern
in_order(a,g)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 CutEliminatorProof (⇒, 0 ms)
↳2 Prolog
↳3 PrologToPiTRSProof (⇒, 64 ms)
↳4 PiTRS
↳5 DependencyPairsProof (⇔, 76 ms)
↳6 PiDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔, 0 ms)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇒, 23 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 MRRProof (⇔, 204 ms)
        ↳20 QDP
        ↳21 PisEmptyProof (⇔, 0 ms)
        ↳22 YES

(0) Obligation:

Clauses:

in_order(void, L) :- ','(!, eq(L, [])).
in_order(T, Xs) :- ','(value(T, X), ','(app(Ls, .(X, Rs), Xs), ','(left(T, L), ','(in_order(L, Ls), ','(right(T, R), in_order(R, Rs)))))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
left(void, void).
left(node(L, X1, X2), L).
right(void, void).
right(node(X3, X4, R), R).
value(void, X5).
value(node(X6, X, X7), X).
eq(X, X).

Query: in_order(a,g)

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

in_order(void, L) :- eq(L, []).
in_order(T, Xs) :- ','(value(T, X), ','(app(Ls, .(X, Rs), Xs), ','(left(T, L), ','(in_order(L, Ls), ','(right(T, R), in_order(R, Rs)))))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
left(void, void).
left(node(L, X1, X2), L).
right(void, void).
right(node(X3, X4, R), R).
value(void, X5).
value(node(X6, X, X7), X).
eq(X, X).

Query: in_order(a,g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_order_in: (f,b)
app_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

in_order_in_ag(void, L) → U1_ag(L, eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg(X, X)
U1_ag(L, eq_out_gg(L, [])) → in_order_out_ag(void, L)
in_order_in_ag(T, Xs) → U2_ag(T, Xs, value_in_aa(T, X))
value_in_aa(void, X5) → value_out_aa(void, X5)
value_in_aa(node(X6, X, X7), X) → value_out_aa(node(X6, X, X7), X)
U2_ag(T, Xs, value_out_aa(T, X)) → U3_ag(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_ag(T, Xs, X, Ls, Rs, left_in_aa(T, L))
left_in_aa(void, void) → left_out_aa(void, void)
left_in_aa(node(L, X1, X2), L) → left_out_aa(node(L, X1, X2), L)
U4_ag(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_ag(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_ag(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_ag(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
right_in_aa(void, void) → right_out_aa(void, void)
right_in_aa(node(X3, X4, R), R) → right_out_aa(node(X3, X4, R), R)
U6_ag(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_ag(T, Xs, in_order_in_ag(R, Rs))
U7_ag(T, Xs, in_order_out_ag(R, Rs)) → in_order_out_ag(T, Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x2)
eq_in_gg(x1, x2)  =  eq_in_gg(x1, x2)
eq_out_gg(x1, x2)  =  eq_out_gg
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x5)
U4_ag(x1, x2, x3, x4, x5, x6)  =  U4_ag(x4, x5, x6)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_ag(x1, x2, x3, x4, x5, x6, x7)  =  U5_ag(x5, x7)
U6_ag(x1, x2, x3, x4, x5, x6, x7)  =  U6_ag(x5, x7)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U7_ag(x1, x2, x3)  =  U7_ag(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

in_order_in_ag(void, L) → U1_ag(L, eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg(X, X)
U1_ag(L, eq_out_gg(L, [])) → in_order_out_ag(void, L)
in_order_in_ag(T, Xs) → U2_ag(T, Xs, value_in_aa(T, X))
value_in_aa(void, X5) → value_out_aa(void, X5)
value_in_aa(node(X6, X, X7), X) → value_out_aa(node(X6, X, X7), X)
U2_ag(T, Xs, value_out_aa(T, X)) → U3_ag(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_ag(T, Xs, X, Ls, Rs, left_in_aa(T, L))
left_in_aa(void, void) → left_out_aa(void, void)
left_in_aa(node(L, X1, X2), L) → left_out_aa(node(L, X1, X2), L)
U4_ag(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_ag(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_ag(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_ag(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
right_in_aa(void, void) → right_out_aa(void, void)
right_in_aa(node(X3, X4, R), R) → right_out_aa(node(X3, X4, R), R)
U6_ag(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_ag(T, Xs, in_order_in_ag(R, Rs))
U7_ag(T, Xs, in_order_out_ag(R, Rs)) → in_order_out_ag(T, Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x2)
eq_in_gg(x1, x2)  =  eq_in_gg(x1, x2)
eq_out_gg(x1, x2)  =  eq_out_gg
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x5)
U4_ag(x1, x2, x3, x4, x5, x6)  =  U4_ag(x4, x5, x6)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_ag(x1, x2, x3, x4, x5, x6, x7)  =  U5_ag(x5, x7)
U6_ag(x1, x2, x3, x4, x5, x6, x7)  =  U6_ag(x5, x7)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U7_ag(x1, x2, x3)  =  U7_ag(x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

IN_ORDER_IN_AG(void, L) → U1_AG(L, eq_in_gg(L, []))
IN_ORDER_IN_AG(void, L) → EQ_IN_GG(L, [])
IN_ORDER_IN_AG(T, Xs) → U2_AG(T, Xs, value_in_aa(T, X))
IN_ORDER_IN_AG(T, Xs) → VALUE_IN_AA(T, X)
U2_AG(T, Xs, value_out_aa(T, X)) → U3_AG(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
U2_AG(T, Xs, value_out_aa(T, X)) → APP_IN_AAG(Ls, .(X, Rs), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U3_AG(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_AG(T, Xs, X, Ls, Rs, left_in_aa(T, L))
U3_AG(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → LEFT_IN_AA(T, L)
U4_AG(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_AG(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U4_AG(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → IN_ORDER_IN_AG(L, Ls)
U5_AG(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_AG(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
U5_AG(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → RIGHT_IN_AA(T, R)
U6_AG(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_AG(T, Xs, in_order_in_ag(R, Rs))
U6_AG(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → IN_ORDER_IN_AG(R, Rs)

The TRS R consists of the following rules:

in_order_in_ag(void, L) → U1_ag(L, eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg(X, X)
U1_ag(L, eq_out_gg(L, [])) → in_order_out_ag(void, L)
in_order_in_ag(T, Xs) → U2_ag(T, Xs, value_in_aa(T, X))
value_in_aa(void, X5) → value_out_aa(void, X5)
value_in_aa(node(X6, X, X7), X) → value_out_aa(node(X6, X, X7), X)
U2_ag(T, Xs, value_out_aa(T, X)) → U3_ag(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_ag(T, Xs, X, Ls, Rs, left_in_aa(T, L))
left_in_aa(void, void) → left_out_aa(void, void)
left_in_aa(node(L, X1, X2), L) → left_out_aa(node(L, X1, X2), L)
U4_ag(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_ag(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_ag(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_ag(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
right_in_aa(void, void) → right_out_aa(void, void)
right_in_aa(node(X3, X4, R), R) → right_out_aa(node(X3, X4, R), R)
U6_ag(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_ag(T, Xs, in_order_in_ag(R, Rs))
U7_ag(T, Xs, in_order_out_ag(R, Rs)) → in_order_out_ag(T, Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x2)
eq_in_gg(x1, x2)  =  eq_in_gg(x1, x2)
eq_out_gg(x1, x2)  =  eq_out_gg
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x5)
U4_ag(x1, x2, x3, x4, x5, x6)  =  U4_ag(x4, x5, x6)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_ag(x1, x2, x3, x4, x5, x6, x7)  =  U5_ag(x5, x7)
U6_ag(x1, x2, x3, x4, x5, x6, x7)  =  U6_ag(x5, x7)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U7_ag(x1, x2, x3)  =  U7_ag(x3)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U1_AG(x1, x2)  =  U1_AG(x2)
EQ_IN_GG(x1, x2)  =  EQ_IN_GG(x1, x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U8_AAG(x1, x2, x3, x4, x5)  =  U8_AAG(x5)
U4_AG(x1, x2, x3, x4, x5, x6)  =  U4_AG(x4, x5, x6)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U5_AG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AG(x5, x7)
U6_AG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AG(x5, x7)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U7_AG(x1, x2, x3)  =  U7_AG(x3)

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

(6) Obligation:

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

IN_ORDER_IN_AG(void, L) → U1_AG(L, eq_in_gg(L, []))
IN_ORDER_IN_AG(void, L) → EQ_IN_GG(L, [])
IN_ORDER_IN_AG(T, Xs) → U2_AG(T, Xs, value_in_aa(T, X))
IN_ORDER_IN_AG(T, Xs) → VALUE_IN_AA(T, X)
U2_AG(T, Xs, value_out_aa(T, X)) → U3_AG(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
U2_AG(T, Xs, value_out_aa(T, X)) → APP_IN_AAG(Ls, .(X, Rs), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U3_AG(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_AG(T, Xs, X, Ls, Rs, left_in_aa(T, L))
U3_AG(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → LEFT_IN_AA(T, L)
U4_AG(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_AG(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U4_AG(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → IN_ORDER_IN_AG(L, Ls)
U5_AG(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_AG(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
U5_AG(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → RIGHT_IN_AA(T, R)
U6_AG(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_AG(T, Xs, in_order_in_ag(R, Rs))
U6_AG(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → IN_ORDER_IN_AG(R, Rs)

The TRS R consists of the following rules:

in_order_in_ag(void, L) → U1_ag(L, eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg(X, X)
U1_ag(L, eq_out_gg(L, [])) → in_order_out_ag(void, L)
in_order_in_ag(T, Xs) → U2_ag(T, Xs, value_in_aa(T, X))
value_in_aa(void, X5) → value_out_aa(void, X5)
value_in_aa(node(X6, X, X7), X) → value_out_aa(node(X6, X, X7), X)
U2_ag(T, Xs, value_out_aa(T, X)) → U3_ag(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_ag(T, Xs, X, Ls, Rs, left_in_aa(T, L))
left_in_aa(void, void) → left_out_aa(void, void)
left_in_aa(node(L, X1, X2), L) → left_out_aa(node(L, X1, X2), L)
U4_ag(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_ag(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_ag(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_ag(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
right_in_aa(void, void) → right_out_aa(void, void)
right_in_aa(node(X3, X4, R), R) → right_out_aa(node(X3, X4, R), R)
U6_ag(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_ag(T, Xs, in_order_in_ag(R, Rs))
U7_ag(T, Xs, in_order_out_ag(R, Rs)) → in_order_out_ag(T, Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x2)
eq_in_gg(x1, x2)  =  eq_in_gg(x1, x2)
eq_out_gg(x1, x2)  =  eq_out_gg
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x5)
U4_ag(x1, x2, x3, x4, x5, x6)  =  U4_ag(x4, x5, x6)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_ag(x1, x2, x3, x4, x5, x6, x7)  =  U5_ag(x5, x7)
U6_ag(x1, x2, x3, x4, x5, x6, x7)  =  U6_ag(x5, x7)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U7_ag(x1, x2, x3)  =  U7_ag(x3)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U1_AG(x1, x2)  =  U1_AG(x2)
EQ_IN_GG(x1, x2)  =  EQ_IN_GG(x1, x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U8_AAG(x1, x2, x3, x4, x5)  =  U8_AAG(x5)
U4_AG(x1, x2, x3, x4, x5, x6)  =  U4_AG(x4, x5, x6)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U5_AG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AG(x5, x7)
U6_AG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AG(x5, x7)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U7_AG(x1, x2, x3)  =  U7_AG(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_in_ag(void, L) → U1_ag(L, eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg(X, X)
U1_ag(L, eq_out_gg(L, [])) → in_order_out_ag(void, L)
in_order_in_ag(T, Xs) → U2_ag(T, Xs, value_in_aa(T, X))
value_in_aa(void, X5) → value_out_aa(void, X5)
value_in_aa(node(X6, X, X7), X) → value_out_aa(node(X6, X, X7), X)
U2_ag(T, Xs, value_out_aa(T, X)) → U3_ag(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_ag(T, Xs, X, Ls, Rs, left_in_aa(T, L))
left_in_aa(void, void) → left_out_aa(void, void)
left_in_aa(node(L, X1, X2), L) → left_out_aa(node(L, X1, X2), L)
U4_ag(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_ag(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_ag(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_ag(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
right_in_aa(void, void) → right_out_aa(void, void)
right_in_aa(node(X3, X4, R), R) → right_out_aa(node(X3, X4, R), R)
U6_ag(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_ag(T, Xs, in_order_in_ag(R, Rs))
U7_ag(T, Xs, in_order_out_ag(R, Rs)) → in_order_out_ag(T, Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x2)
eq_in_gg(x1, x2)  =  eq_in_gg(x1, x2)
eq_out_gg(x1, x2)  =  eq_out_gg
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x5)
U4_ag(x1, x2, x3, x4, x5, x6)  =  U4_ag(x4, x5, x6)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_ag(x1, x2, x3, x4, x5, x6, x7)  =  U5_ag(x5, x7)
U6_ag(x1, x2, x3, x4, x5, x6, x7)  =  U6_ag(x5, x7)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U7_ag(x1, x2, x3)  =  U7_ag(x3)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP_IN_AAG(.(Zs)) → APP_IN_AAG(Zs)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • APP_IN_AAG(.(Zs)) → APP_IN_AAG(Zs)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

IN_ORDER_IN_AG(T, Xs) → U2_AG(T, Xs, value_in_aa(T, X))
U2_AG(T, Xs, value_out_aa(T, X)) → U3_AG(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
U3_AG(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_AG(T, Xs, X, Ls, Rs, left_in_aa(T, L))
U4_AG(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_AG(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_AG(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_AG(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
U6_AG(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → IN_ORDER_IN_AG(R, Rs)
U4_AG(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → IN_ORDER_IN_AG(L, Ls)

The TRS R consists of the following rules:

in_order_in_ag(void, L) → U1_ag(L, eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg(X, X)
U1_ag(L, eq_out_gg(L, [])) → in_order_out_ag(void, L)
in_order_in_ag(T, Xs) → U2_ag(T, Xs, value_in_aa(T, X))
value_in_aa(void, X5) → value_out_aa(void, X5)
value_in_aa(node(X6, X, X7), X) → value_out_aa(node(X6, X, X7), X)
U2_ag(T, Xs, value_out_aa(T, X)) → U3_ag(T, Xs, X, app_in_aag(Ls, .(X, Rs), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(T, Xs, X, app_out_aag(Ls, .(X, Rs), Xs)) → U4_ag(T, Xs, X, Ls, Rs, left_in_aa(T, L))
left_in_aa(void, void) → left_out_aa(void, void)
left_in_aa(node(L, X1, X2), L) → left_out_aa(node(L, X1, X2), L)
U4_ag(T, Xs, X, Ls, Rs, left_out_aa(T, L)) → U5_ag(T, Xs, X, Ls, Rs, L, in_order_in_ag(L, Ls))
U5_ag(T, Xs, X, Ls, Rs, L, in_order_out_ag(L, Ls)) → U6_ag(T, Xs, X, Ls, Rs, L, right_in_aa(T, R))
right_in_aa(void, void) → right_out_aa(void, void)
right_in_aa(node(X3, X4, R), R) → right_out_aa(node(X3, X4, R), R)
U6_ag(T, Xs, X, Ls, Rs, L, right_out_aa(T, R)) → U7_ag(T, Xs, in_order_in_ag(R, Rs))
U7_ag(T, Xs, in_order_out_ag(R, Rs)) → in_order_out_ag(T, Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x2)
eq_in_gg(x1, x2)  =  eq_in_gg(x1, x2)
eq_out_gg(x1, x2)  =  eq_out_gg
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x5)
U4_ag(x1, x2, x3, x4, x5, x6)  =  U4_ag(x4, x5, x6)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_ag(x1, x2, x3, x4, x5, x6, x7)  =  U5_ag(x5, x7)
U6_ag(x1, x2, x3, x4, x5, x6, x7)  =  U6_ag(x5, x7)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U7_ag(x1, x2, x3)  =  U7_ag(x3)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
U4_AG(x1, x2, x3, x4, x5, x6)  =  U4_AG(x4, x5, x6)
U5_AG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AG(x5, x7)
U6_AG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AG(x5, x7)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

IN_ORDER_IN_AG(Xs) → U2_AG(Xs, value_in_aa)
U2_AG(Xs, value_out_aa) → U3_AG(app_in_aag(Xs))
U3_AG(app_out_aag(Ls, .(Rs))) → U4_AG(Ls, Rs, left_in_aa)
U4_AG(Ls, Rs, left_out_aa) → U5_AG(Rs, in_order_in_ag(Ls))
U5_AG(Rs, in_order_out_ag) → U6_AG(Rs, right_in_aa)
U6_AG(Rs, right_out_aa) → IN_ORDER_IN_AG(Rs)
U4_AG(Ls, Rs, left_out_aa) → IN_ORDER_IN_AG(Ls)

The TRS R consists of the following rules:

in_order_in_ag(L) → U1_ag(eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg
U1_ag(eq_out_gg) → in_order_out_ag
in_order_in_ag(Xs) → U2_ag(Xs, value_in_aa)
value_in_aa → value_out_aa
U2_ag(Xs, value_out_aa) → U3_ag(app_in_aag(Xs))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(Zs)) → U8_aag(app_in_aag(Zs))
U8_aag(app_out_aag(Xs, Ys)) → app_out_aag(.(Xs), Ys)
U3_ag(app_out_aag(Ls, .(Rs))) → U4_ag(Ls, Rs, left_in_aa)
left_in_aa → left_out_aa
U4_ag(Ls, Rs, left_out_aa) → U5_ag(Rs, in_order_in_ag(Ls))
U5_ag(Rs, in_order_out_ag) → U6_ag(Rs, right_in_aa)
right_in_aa → right_out_aa
U6_ag(Rs, right_out_aa) → U7_ag(in_order_in_ag(Rs))
U7_ag(in_order_out_ag) → in_order_out_ag

The set Q consists of the following terms:

in_order_in_ag(x0)
eq_in_gg(x0, x1)
U1_ag(x0)
value_in_aa
U2_ag(x0, x1)
app_in_aag(x0)
U8_aag(x0)
U3_ag(x0)
left_in_aa
U4_ag(x0, x1, x2)
U5_ag(x0, x1)
right_in_aa
U6_ag(x0, x1)
U7_ag(x0)

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

(19) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

IN_ORDER_IN_AG(Xs) → U2_AG(Xs, value_in_aa)
U2_AG(Xs, value_out_aa) → U3_AG(app_in_aag(Xs))
U3_AG(app_out_aag(Ls, .(Rs))) → U4_AG(Ls, Rs, left_in_aa)
U4_AG(Ls, Rs, left_out_aa) → U5_AG(Rs, in_order_in_ag(Ls))
U5_AG(Rs, in_order_out_ag) → U6_AG(Rs, right_in_aa)
U6_AG(Rs, right_out_aa) → IN_ORDER_IN_AG(Rs)
U4_AG(Ls, Rs, left_out_aa) → IN_ORDER_IN_AG(Ls)

Strictly oriented rules of the TRS R:

in_order_in_ag(L) → U1_ag(eq_in_gg(L, []))
eq_in_gg(X, X) → eq_out_gg
U1_ag(eq_out_gg) → in_order_out_ag
in_order_in_ag(Xs) → U2_ag(Xs, value_in_aa)
value_in_aa → value_out_aa
U2_ag(Xs, value_out_aa) → U3_ag(app_in_aag(Xs))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(Zs)) → U8_aag(app_in_aag(Zs))
U8_aag(app_out_aag(Xs, Ys)) → app_out_aag(.(Xs), Ys)
U3_ag(app_out_aag(Ls, .(Rs))) → U4_ag(Ls, Rs, left_in_aa)
left_in_aa → left_out_aa
U4_ag(Ls, Rs, left_out_aa) → U5_ag(Rs, in_order_in_ag(Ls))
U5_ag(Rs, in_order_out_ag) → U6_ag(Rs, right_in_aa)
right_in_aa → right_out_aa
U6_ag(Rs, right_out_aa) → U7_ag(in_order_in_ag(Rs))
U7_ag(in_order_out_ag) → in_order_out_ag

Used ordering: Knuth-Bendix order [KBO] with precedence:
inorderinag1 > eqoutgg > U2ag2 > U3AG1 > U4AG3 > rightinaa > appinaag1 > U5AG2 > U6AG2 > INORDERINAG1 > U2AG2 > leftinaa > valueinaa > rightoutaa > U6ag2 > eqingg2 > leftoutaa > U3ag1 > U7ag1 > U8aag1 > U4ag3 > U5ag2 > U1ag1 > inorderoutag > appoutaag2 > valueoutaa > .1 > []

and weight map:

[]=7
eq_out_gg=11
in_order_out_ag=18
value_in_aa=7
value_out_aa=6
left_in_aa=18
left_out_aa=17
right_in_aa=19
right_out_aa=17
in_order_in_ag_1=15
U1_ag_1=7
U3_ag_1=1
app_in_aag_1=13
._1=12
U8_aag_1=12
U7_ag_1=1
IN_ORDER_IN_AG_1=17
U3_AG_1=1
eq_in_gg_2=0
U2_ag_2=8
app_out_aag_2=5
U4_ag_3=0
U5_ag_2=2
U6_ag_2=0
U2_AG_2=10
U4_AG_3=0
U5_AG_2=2
U6_AG_2=0

The variable weight is 6

(20) Obligation:

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

in_order_in_ag(x0)
eq_in_gg(x0, x1)
U1_ag(x0)
value_in_aa
U2_ag(x0, x1)
app_in_aag(x0)
U8_aag(x0)
U3_ag(x0)
left_in_aa
U4_ag(x0, x1, x2)
U5_ag(x0, x1)
right_in_aa
U6_ag(x0, x1)
U7_ag(x0)

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

(21) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(22) YES