Left Termination of the query pattern in_order_in_2(a, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

in_order(void, []).
in_order(tree(X, Left, Right), Xs) :- ','(in_order(Left, Ls), ','(in_order(Right, Rs), app(Ls, .(X, Rs), Xs))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

in_order(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_order_in: (f,b) (f,f)
app_in: (b,b,f) (b,b,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, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)

(3) 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(tree(X, Left, Right), Xs) → U1_AG(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AG(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AA(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGA(Ls, .(X, Rs), Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AG(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AG(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGG(Ls, .(X, Rs), Xs)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x2, x5, x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x2, x4, x5, x6)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x5)
APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4, x5)  =  U4_GGG(x5)

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

(4) Obligation:

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

IN_ORDER_IN_AG(tree(X, Left, Right), Xs) → U1_AG(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AG(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AA(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGA(Ls, .(X, Rs), Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AG(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AG(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGG(Ls, .(X, Rs), Xs)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x2, x5, x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x2, x4, x5, x6)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x5)
APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4, x5)  =  U4_GGG(x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)
APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APP_IN_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)

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

(12) 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_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(13) TRUE

(14) Obligation:

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

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

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

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

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:

APP_IN_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)

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

(19) 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_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) TRUE

(21) Obligation:

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

U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x5)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)

The TRS R consists of the following rules:

in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
tree(x1, x2, x3)  =  tree(x2, x3)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U1_AA(in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA
IN_ORDER_IN_AA → U1_AA(in_order_in_aa)
IN_ORDER_IN_AA → IN_ORDER_IN_AA

The TRS R consists of the following rules:

in_order_in_aa → in_order_out_aa(void, [])
in_order_in_aa → U1_aa(in_order_in_aa)
U1_aa(in_order_out_aa(Left, Ls)) → U2_aa(Left, Ls, in_order_in_aa)
U2_aa(Left, Ls, in_order_out_aa(Right, Rs)) → U3_aa(Left, Right, app_in_gga(Ls, .(Rs)))
U3_aa(Left, Right, app_out_gga(Xs)) → in_order_out_aa(tree(Left, Right), Xs)
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(Xs), Ys) → U4_gga(app_in_gga(Xs, Ys))
U4_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))

The set Q consists of the following terms:

in_order_in_aa
U1_aa(x0)
U2_aa(x0, x1, x2)
U3_aa(x0, x1, x2)
app_in_gga(x0, x1)
U4_gga(x0)

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

(26) Narrowing (SOUND transformation)

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

IN_ORDER_IN_AA → U1_AA(in_order_out_aa(void, []))
IN_ORDER_IN_AA → U1_AA(U1_aa(in_order_in_aa))

(27) Obligation:

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

U1_AA(in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA
IN_ORDER_IN_AA → IN_ORDER_IN_AA
IN_ORDER_IN_AA → U1_AA(in_order_out_aa(void, []))
IN_ORDER_IN_AA → U1_AA(U1_aa(in_order_in_aa))

The TRS R consists of the following rules:

in_order_in_aa → in_order_out_aa(void, [])
in_order_in_aa → U1_aa(in_order_in_aa)
U1_aa(in_order_out_aa(Left, Ls)) → U2_aa(Left, Ls, in_order_in_aa)
U2_aa(Left, Ls, in_order_out_aa(Right, Rs)) → U3_aa(Left, Right, app_in_gga(Ls, .(Rs)))
U3_aa(Left, Right, app_out_gga(Xs)) → in_order_out_aa(tree(Left, Right), Xs)
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(Xs), Ys) → U4_gga(app_in_gga(Xs, Ys))
U4_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))

The set Q consists of the following terms:

in_order_in_aa
U1_aa(x0)
U2_aa(x0, x1, x2)
U3_aa(x0, x1, x2)
app_in_gga(x0, x1)
U4_gga(x0)

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

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = IN_ORDER_IN_AA evaluates to t =IN_ORDER_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from IN_ORDER_IN_AA to IN_ORDER_IN_AA.

(29) FALSE

(30) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_order_in: (f,b) (f,f)
app_in: (b,b,f) (b,b,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, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

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

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)

(32) 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(tree(X, Left, Right), Xs) → U1_AG(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AG(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AA(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGA(Ls, .(X, Rs), Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AG(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AG(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGG(Ls, .(X, Rs), Xs)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x2, x5, x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x3, x5)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x2, x4, x5, x6)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x4, x5)
APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4, x5)  =  U4_GGG(x2, x3, x4, x5)

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

(33) Obligation:

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

IN_ORDER_IN_AG(tree(X, Left, Right), Xs) → U1_AG(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AG(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AA(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
U2_AA(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGA(Ls, .(X, Rs), Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_AG(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U1_AG(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_AG(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
U2_AG(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → APP_IN_GGG(Ls, .(X, Rs), Xs)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)
IN_ORDER_IN_AG(x1, x2)  =  IN_ORDER_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x2, x5, x6)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x3, x5)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x2, x4, x5, x6)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x4, x5)
APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4, x5)  =  U4_GGG(x2, x3, x4, x5)

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

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(35) Complex Obligation (AND)

(36) Obligation:

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

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

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)
APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

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

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

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

APP_IN_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)

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

(41) 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_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(42) TRUE

(43) Obligation:

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

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

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

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

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

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

APP_IN_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)

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

(48) 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_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)
  The graph contains the following edges 1 > 1, 2 >= 2

(49) TRUE

(50) Obligation:

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

U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)

The TRS R consists of the following rules:

in_order_in_ag(void, []) → in_order_out_ag(void, [])
in_order_in_ag(tree(X, Left, Right), Xs) → U1_ag(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
U1_ag(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_ag(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_ag(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_ag(X, Left, Right, Xs, app_in_ggg(Ls, .(X, Rs), Xs))
app_in_ggg([], X, X) → app_out_ggg([], X, X)
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Left, Right, Xs, app_out_ggg(Ls, .(X, Rs), Xs)) → in_order_out_ag(tree(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_in_ag(x1, x2)  =  in_order_in_ag(x2)
[]  =  []
in_order_out_ag(x1, x2)  =  in_order_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x2, x4, x5, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5)  =  U4_ggg(x2, x3, x4, x5)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

U1_AA(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA(Right, Rs)
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → U1_AA(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
IN_ORDER_IN_AA(tree(X, Left, Right), Xs) → IN_ORDER_IN_AA(Left, Ls)

The TRS R consists of the following rules:

in_order_in_aa(void, []) → in_order_out_aa(void, [])
in_order_in_aa(tree(X, Left, Right), Xs) → U1_aa(X, Left, Right, Xs, in_order_in_aa(Left, Ls))
U1_aa(X, Left, Right, Xs, in_order_out_aa(Left, Ls)) → U2_aa(X, Left, Right, Xs, Ls, in_order_in_aa(Right, Rs))
U2_aa(X, Left, Right, Xs, Ls, in_order_out_aa(Right, Rs)) → U3_aa(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs))
U3_aa(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) → in_order_out_aa(tree(X, Left, Right), Xs)
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
in_order_in_aa(x1, x2)  =  in_order_in_aa
in_order_out_aa(x1, x2)  =  in_order_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x2, x5, x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
.(x1, x2)  =  .(x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
tree(x1, x2, x3)  =  tree(x2, x3)
IN_ORDER_IN_AA(x1, x2)  =  IN_ORDER_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

U1_AA(in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA
IN_ORDER_IN_AA → U1_AA(in_order_in_aa)
IN_ORDER_IN_AA → IN_ORDER_IN_AA

The TRS R consists of the following rules:

in_order_in_aa → in_order_out_aa(void, [])
in_order_in_aa → U1_aa(in_order_in_aa)
U1_aa(in_order_out_aa(Left, Ls)) → U2_aa(Left, Ls, in_order_in_aa)
U2_aa(Left, Ls, in_order_out_aa(Right, Rs)) → U3_aa(Left, Right, app_in_gga(Ls, .(Rs)))
U3_aa(Left, Right, app_out_gga(Ls, .(Rs), Xs)) → in_order_out_aa(tree(Left, Right), Xs)
app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(.(Xs), Ys) → U4_gga(Xs, Ys, app_in_gga(Xs, Ys))
U4_gga(Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(Xs), Ys, .(Zs))

The set Q consists of the following terms:

in_order_in_aa
U1_aa(x0)
U2_aa(x0, x1, x2)
U3_aa(x0, x1, x2)
app_in_gga(x0, x1)
U4_gga(x0, x1, x2)

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

(55) Narrowing (SOUND transformation)

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

IN_ORDER_IN_AA → U1_AA(in_order_out_aa(void, []))
IN_ORDER_IN_AA → U1_AA(U1_aa(in_order_in_aa))

(56) Obligation:

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

U1_AA(in_order_out_aa(Left, Ls)) → IN_ORDER_IN_AA
IN_ORDER_IN_AA → IN_ORDER_IN_AA
IN_ORDER_IN_AA → U1_AA(in_order_out_aa(void, []))
IN_ORDER_IN_AA → U1_AA(U1_aa(in_order_in_aa))

The TRS R consists of the following rules:

in_order_in_aa → in_order_out_aa(void, [])
in_order_in_aa → U1_aa(in_order_in_aa)
U1_aa(in_order_out_aa(Left, Ls)) → U2_aa(Left, Ls, in_order_in_aa)
U2_aa(Left, Ls, in_order_out_aa(Right, Rs)) → U3_aa(Left, Right, app_in_gga(Ls, .(Rs)))
U3_aa(Left, Right, app_out_gga(Ls, .(Rs), Xs)) → in_order_out_aa(tree(Left, Right), Xs)
app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(.(Xs), Ys) → U4_gga(Xs, Ys, app_in_gga(Xs, Ys))
U4_gga(Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(Xs), Ys, .(Zs))

The set Q consists of the following terms:

in_order_in_aa
U1_aa(x0)
U2_aa(x0, x1, x2)
U3_aa(x0, x1, x2)
app_in_gga(x0, x1)
U4_gga(x0, x1, x2)

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

(57) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = IN_ORDER_IN_AA evaluates to t =IN_ORDER_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

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Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from IN_ORDER_IN_AA to IN_ORDER_IN_AA.

(58) FALSE