Left Termination of the query pattern front_in_2(a, g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

front(void, []).
front(tree(X, void, void), .(X, [])).
front(tree(X, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

front(a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
front_in: (f,b) (f,f)
app_in: (f,f,f) (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FRONT_IN_AG(tree(X, L, R), Xs) → U1_AG(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AG(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AA(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AA(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAA(Ls, Rs, Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AG(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AG(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAG(Ls, Rs, Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x1, x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
FRONT_IN_AG(x1, x2)  =  FRONT_IN_AG(x2)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

FRONT_IN_AG(tree(X, L, R), Xs) → U1_AG(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AG(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AA(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AA(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAA(Ls, Rs, Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AG(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AG(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAG(Ls, Rs, Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x1, x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
FRONT_IN_AG(x1, x2)  =  FRONT_IN_AG(x2)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 11 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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)  =  .(x1, x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

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

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_AAA → APP_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none


s = APP_IN_AAA evaluates to t =APP_IN_AAA

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



Rewriting sequence

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





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)

The TRS R consists of the following rules:

front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

FRONT_IN_AA → U1_AA(front_in_aa)
U1_AA(front_out_aa) → FRONT_IN_AA
FRONT_IN_AA → FRONT_IN_AA

The TRS R consists of the following rules:

front_in_aa → front_out_aa
front_in_aa → U1_aa(front_in_aa)
U1_aa(front_out_aa) → U2_aa(front_in_aa)
U2_aa(front_out_aa) → U3_aa(app_in_aaa)
U3_aa(app_out_aaa) → front_out_aa
app_in_aaa → app_out_aaa
app_in_aaa → U4_aaa(app_in_aaa)
U4_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

front_in_aa
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
app_in_aaa
U4_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule FRONT_IN_AA → U1_AA(front_in_aa) at position [0] we obtained the following new rules:

FRONT_IN_AA → U1_AA(U1_aa(front_in_aa))
FRONT_IN_AA → U1_AA(front_out_aa)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

FRONT_IN_AA → U1_AA(U1_aa(front_in_aa))
FRONT_IN_AA → U1_AA(front_out_aa)
FRONT_IN_AA → FRONT_IN_AA
U1_AA(front_out_aa) → FRONT_IN_AA

The TRS R consists of the following rules:

front_in_aa → front_out_aa
front_in_aa → U1_aa(front_in_aa)
U1_aa(front_out_aa) → U2_aa(front_in_aa)
U2_aa(front_out_aa) → U3_aa(app_in_aaa)
U3_aa(app_out_aaa) → front_out_aa
app_in_aaa → app_out_aaa
app_in_aaa → U4_aaa(app_in_aaa)
U4_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

front_in_aa
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
app_in_aaa
U4_aaa(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

FRONT_IN_AA → U1_AA(U1_aa(front_in_aa))
FRONT_IN_AA → U1_AA(front_out_aa)
FRONT_IN_AA → FRONT_IN_AA
U1_AA(front_out_aa) → FRONT_IN_AA

The TRS R consists of the following rules:

front_in_aa → front_out_aa
front_in_aa → U1_aa(front_in_aa)
U1_aa(front_out_aa) → U2_aa(front_in_aa)
U2_aa(front_out_aa) → U3_aa(app_in_aaa)
U3_aa(app_out_aaa) → front_out_aa
app_in_aaa → app_out_aaa
app_in_aaa → U4_aaa(app_in_aaa)
U4_aaa(app_out_aaa) → app_out_aaa


s = FRONT_IN_AA evaluates to t =FRONT_IN_AA

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



Rewriting sequence

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




We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
front_in: (f,b) (f,f)
app_in: (f,f,f) (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FRONT_IN_AG(tree(X, L, R), Xs) → U1_AG(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AG(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AA(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AA(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAA(Ls, Rs, Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AG(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AG(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAG(Ls, Rs, Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x1, x4, x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
FRONT_IN_AG(x1, x2)  =  FRONT_IN_AG(x2)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

FRONT_IN_AG(tree(X, L, R), Xs) → U1_AG(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AG(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AA(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AA(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
U2_AA(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAA(Ls, Rs, Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U4_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → U2_AG(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U1_AG(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_AG(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
U2_AG(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → APP_IN_AAG(Ls, Rs, Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x1, x4, x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
FRONT_IN_AG(x1, x2)  =  FRONT_IN_AG(x2)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 11 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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)  =  .(x1, x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

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

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

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

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

APP_IN_AAA → APP_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none


s = APP_IN_AAA evaluates to t =APP_IN_AAA

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



Rewriting sequence

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





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)

The TRS R consists of the following rules:

front_in_ag(void, []) → front_out_ag(void, [])
front_in_ag(tree(X, void, void), .(X, [])) → front_out_ag(tree(X, void, void), .(X, []))
front_in_ag(tree(X, L, R), Xs) → U1_ag(X, L, R, Xs, front_in_aa(L, Ls))
front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
U1_ag(X, L, R, Xs, front_out_aa(L, Ls)) → U2_ag(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_ag(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_ag(X, L, R, Xs, app_in_aag(Ls, Rs, Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, L, R, Xs, app_out_aag(Ls, Rs, Xs)) → front_out_ag(tree(X, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ag(x1, x2)  =  front_in_ag(x2)
[]  =  []
front_out_ag(x1, x2)  =  front_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x1, x4, x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

FRONT_IN_AA(tree(X, L, R), Xs) → U1_AA(X, L, R, Xs, front_in_aa(L, Ls))
FRONT_IN_AA(tree(X, L, R), Xs) → FRONT_IN_AA(L, Ls)
U1_AA(X, L, R, Xs, front_out_aa(L, Ls)) → FRONT_IN_AA(R, Rs)

The TRS R consists of the following rules:

front_in_aa(void, []) → front_out_aa(void, [])
front_in_aa(tree(X, void, void), .(X, [])) → front_out_aa(tree(X, void, void), .(X, []))
front_in_aa(tree(X, L, R), Xs) → U1_aa(X, L, R, Xs, front_in_aa(L, Ls))
U1_aa(X, L, R, Xs, front_out_aa(L, Ls)) → U2_aa(X, L, R, Xs, Ls, front_in_aa(R, Rs))
U2_aa(X, L, R, Xs, Ls, front_out_aa(R, Rs)) → U3_aa(X, L, R, Xs, app_in_aaa(Ls, Rs, Xs))
U3_aa(X, L, R, Xs, app_out_aaa(Ls, Rs, Xs)) → front_out_aa(tree(X, L, R), Xs)
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U4_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U4_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
front_in_aa(x1, x2)  =  front_in_aa
front_out_aa(x1, x2)  =  front_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
FRONT_IN_AA(x1, x2)  =  FRONT_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing

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

FRONT_IN_AA → U1_AA(front_in_aa)
U1_AA(front_out_aa) → FRONT_IN_AA
FRONT_IN_AA → FRONT_IN_AA

The TRS R consists of the following rules:

front_in_aa → front_out_aa
front_in_aa → U1_aa(front_in_aa)
U1_aa(front_out_aa) → U2_aa(front_in_aa)
U2_aa(front_out_aa) → U3_aa(app_in_aaa)
U3_aa(app_out_aaa) → front_out_aa
app_in_aaa → app_out_aaa
app_in_aaa → U4_aaa(app_in_aaa)
U4_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

front_in_aa
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
app_in_aaa
U4_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule FRONT_IN_AA → U1_AA(front_in_aa) at position [0] we obtained the following new rules:

FRONT_IN_AA → U1_AA(U1_aa(front_in_aa))
FRONT_IN_AA → U1_AA(front_out_aa)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ NonTerminationProof

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

FRONT_IN_AA → U1_AA(U1_aa(front_in_aa))
FRONT_IN_AA → U1_AA(front_out_aa)
FRONT_IN_AA → FRONT_IN_AA
U1_AA(front_out_aa) → FRONT_IN_AA

The TRS R consists of the following rules:

front_in_aa → front_out_aa
front_in_aa → U1_aa(front_in_aa)
U1_aa(front_out_aa) → U2_aa(front_in_aa)
U2_aa(front_out_aa) → U3_aa(app_in_aaa)
U3_aa(app_out_aaa) → front_out_aa
app_in_aaa → app_out_aaa
app_in_aaa → U4_aaa(app_in_aaa)
U4_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

front_in_aa
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
app_in_aaa
U4_aaa(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

FRONT_IN_AA → U1_AA(U1_aa(front_in_aa))
FRONT_IN_AA → U1_AA(front_out_aa)
FRONT_IN_AA → FRONT_IN_AA
U1_AA(front_out_aa) → FRONT_IN_AA

The TRS R consists of the following rules:

front_in_aa → front_out_aa
front_in_aa → U1_aa(front_in_aa)
U1_aa(front_out_aa) → U2_aa(front_in_aa)
U2_aa(front_out_aa) → U3_aa(app_in_aaa)
U3_aa(app_out_aaa) → front_out_aa
app_in_aaa → app_out_aaa
app_in_aaa → U4_aaa(app_in_aaa)
U4_aaa(app_out_aaa) → app_out_aaa


s = FRONT_IN_AA evaluates to t =FRONT_IN_AA

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



Rewriting sequence

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