Left Termination proof of /tmp/tmp9fWCiN/perm.pl

Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

append(nil, XS, XS).
append(cons(X, XS1), XS2, cons(X, YS)) :- append(XS1, XS2, YS).
split(XS, nil, XS).
split(cons(X, XS), cons(X, YS1), YS2) :- split(XS, YS1, YS2).
perm(nil, nil).
perm(XS, cons(Y, YS)) :- ','(split(XS, YS1, cons(Y, YS2)), ','(append(YS1, YS2, ZS), perm(ZS, YS))).

Queries:

perm(g,a).

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

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

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:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAG(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → U2_GAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → U2_AAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → U1_GAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → U1_AAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
PERM_IN_AA(XS, cons(Y, YS)) → SPLIT_IN_AAG(XS, YS1, cons(Y, YS2))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_AA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
nil = nil
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
split_in_gag(x1, x2, x3) = split_in_gag(x1, x3)
cons(x1, x2) = cons
split_out_gag(x1, x2, x3) = split_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
split_in_aag(x1, x2, x3) = split_in_aag(x3)
split_out_aag(x1, x2, x3) = split_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa(x1)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
perm_in_aa(x1, x2) = perm_in_aa
perm_out_aa(x1, x2) = perm_out_aa(x1, x2)
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6) = U4_aa(x1, x6)
U5_aa(x1, x2, x3, x4) = U5_aa(x1, x4)
U5_AA(x1, x2, x3, x4) = U5_AA(x1, x4)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
SPLIT_IN_AAG(x1, x2, x3) = SPLIT_IN_AAG(x3)
U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x1, x6)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5)
U5_GA(x1, x2, x3, x4) = U5_GA(x4)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6)
PERM_IN_AA(x1, x2) = PERM_IN_AA
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)
SPLIT_IN_GAG(x1, x2, x3) = SPLIT_IN_GAG(x1, x3)
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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:

PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAG(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → U2_GAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → U2_AAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → U1_GAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → U1_AAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
PERM_IN_AA(XS, cons(Y, YS)) → SPLIT_IN_AAG(XS, YS1, cons(Y, YS2))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_AA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

↳ 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:

APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APPEND_IN_AAA → APPEND_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:

APPEND_IN_AAA → APPEND_IN_AAA

The TRS R consists of the following rules:none

s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA

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 APPEND_IN_AAA to APPEND_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

↳ 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:

SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

SPLIT_IN_AAG(YS2) → SPLIT_IN_AAG(YS2)

The TRS R consists of the following rules:none

s = SPLIT_IN_AAG(YS2) evaluates to t =SPLIT_IN_AAG(YS2)

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 SPLIT_IN_AAG(YS2) to SPLIT_IN_AAG(YS2).

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
nil = nil
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
split_in_gag(x1, x2, x3) = split_in_gag(x1, x3)
cons(x1, x2) = cons
split_out_gag(x1, x2, x3) = split_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
split_in_aag(x1, x2, x3) = split_in_aag(x3)
split_out_aag(x1, x2, x3) = split_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa(x1)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
perm_in_aa(x1, x2) = perm_in_aa
perm_out_aa(x1, x2) = perm_out_aa(x1, x2)
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6) = U4_aa(x1, x6)
U5_aa(x1, x2, x3, x4) = U5_aa(x1, x4)
U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x1, x6)
PERM_IN_AA(x1, x2) = PERM_IN_AA
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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:

U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))

The argument filtering Pi contains the following mapping:
nil = nil
cons(x1, x2) = cons
split_in_aag(x1, x2, x3) = split_in_aag(x3)
split_out_aag(x1, x2, x3) = split_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa(x1)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x1, x6)
PERM_IN_AA(x1, x2) = PERM_IN_AA
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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:

U3_AA(split_out_aag(XS, YS1)) → U4_AA(XS, append_in_gaa(YS1))
PERM_IN_AA → U3_AA(split_in_aag(cons))
U4_AA(XS, append_out_gaa) → PERM_IN_AA

The TRS R consists of the following rules:

append_in_gaa(nil) → append_out_gaa
append_in_gaa(cons) → U1_gaa(append_in_aaa)
split_in_aag(XS) → split_out_aag(XS, nil)
split_in_aag(YS2) → U2_aag(split_in_aag(YS2))
U1_gaa(append_out_aaa(XS1)) → append_out_gaa
U2_aag(split_out_aag(XS, YS1)) → split_out_aag(cons, cons)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

append_in_gaa(x0)
split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U3_AA(split_out_aag(XS, YS1)) → U4_AA(XS, append_in_gaa(YS1)) at position [1] we obtained the following new rules:

U3_AA(split_out_aag(y0, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))
U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U3_AA(split_out_aag(y0, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))
U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)
PERM_IN_AA → U3_AA(split_in_aag(cons))
U4_AA(XS, append_out_gaa) → PERM_IN_AA

The TRS R consists of the following rules:

append_in_gaa(nil) → append_out_gaa
append_in_gaa(cons) → U1_gaa(append_in_aaa)
split_in_aag(XS) → split_out_aag(XS, nil)
split_in_aag(YS2) → U2_aag(split_in_aag(YS2))
U1_gaa(append_out_aaa(XS1)) → append_out_gaa
U2_aag(split_out_aag(XS, YS1)) → split_out_aag(cons, cons)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

append_in_gaa(x0)
split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

  ↳ PrologToPiTRSProof

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

U3_AA(split_out_aag(y0, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))
U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)
PERM_IN_AA → U3_AA(split_in_aag(cons))
U4_AA(XS, append_out_gaa) → PERM_IN_AA

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil)
split_in_aag(YS2) → U2_aag(split_in_aag(YS2))
U2_aag(split_out_aag(XS, YS1)) → split_out_aag(cons, cons)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

append_in_gaa(x0)
split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

append_in_gaa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

  ↳ PrologToPiTRSProof

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

U3_AA(split_out_aag(y0, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))
U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)
PERM_IN_AA → U3_AA(split_in_aag(cons))
U4_AA(XS, append_out_gaa) → PERM_IN_AA

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil)
split_in_aag(YS2) → U2_aag(split_in_aag(YS2))
U2_aag(split_out_aag(XS, YS1)) → split_out_aag(cons, cons)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PERM_IN_AA → U3_AA(split_in_aag(cons)) at position [0] we obtained the following new rules:

PERM_IN_AA → U3_AA(split_out_aag(cons, nil))
PERM_IN_AA → U3_AA(U2_aag(split_in_aag(cons)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

U3_AA(split_out_aag(y0, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))
U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)
PERM_IN_AA → U3_AA(split_out_aag(cons, nil))
PERM_IN_AA → U3_AA(U2_aag(split_in_aag(cons)))
U4_AA(XS, append_out_gaa) → PERM_IN_AA

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil)
split_in_aag(YS2) → U2_aag(split_in_aag(YS2))
U2_aag(split_out_aag(XS, YS1)) → split_out_aag(cons, cons)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0)
append_in_aaa
U1_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 narrowing to the left:

The TRS P consists of the following rules:

U3_AA(split_out_aag(y0, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))
U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)
PERM_IN_AA → U3_AA(split_out_aag(cons, nil))
PERM_IN_AA → U3_AA(U2_aag(split_in_aag(cons)))
U4_AA(XS, append_out_gaa) → PERM_IN_AA

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil)
split_in_aag(YS2) → U2_aag(split_in_aag(YS2))
U2_aag(split_out_aag(XS, YS1)) → split_out_aag(cons, cons)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

s = U4_AA(XS, append_out_gaa) evaluates to t =U4_AA(cons, append_out_gaa)

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

Matcher: [XS / cons]
Semiunifier: [ ]

Rewriting sequence

U4_AA(XS, append_out_gaa) → PERM_IN_AA
with rule U4_AA(XS', append_out_gaa) → PERM_IN_AA at position [] and matcher [XS' / XS]

PERM_IN_AA → U3_AA(split_out_aag(cons, nil))
with rule PERM_IN_AA → U3_AA(split_out_aag(cons, nil)) at position [] and matcher [ ]

U3_AA(split_out_aag(cons, nil)) → U4_AA(cons, append_out_gaa)
with rule U3_AA(split_out_aag(y0, nil)) → U4_AA(y0, append_out_gaa)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

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

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

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:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAG(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → U2_GAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → U2_AAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → U1_GAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → U1_AAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
PERM_IN_AA(XS, cons(Y, YS)) → SPLIT_IN_AAG(XS, YS1, cons(Y, YS2))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_AA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
nil = nil
perm_out_ga(x1, x2) = perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4)
split_in_gag(x1, x2, x3) = split_in_gag(x1, x3)
cons(x1, x2) = cons
split_out_gag(x1, x2, x3) = split_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x4, x5)
split_in_aag(x1, x2, x3) = split_in_aag(x3)
split_out_aag(x1, x2, x3) = split_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x1, x6)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
append_out_gaa(x1, x2, x3) = append_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa(x1)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4)
perm_in_aa(x1, x2) = perm_in_aa
perm_out_aa(x1, x2) = perm_out_aa(x1, x2)
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6) = U4_aa(x1, x6)
U5_aa(x1, x2, x3, x4) = U5_aa(x1, x4)
U5_AA(x1, x2, x3, x4) = U5_AA(x1, x4)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
SPLIT_IN_AAG(x1, x2, x3) = SPLIT_IN_AAG(x3)
U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x1, x6)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x4, x5)
U3_GA(x1, x2, x3, x4) = U3_GA(x1, x4)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5)
U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x1, x6)
PERM_IN_AA(x1, x2) = PERM_IN_AA
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)
SPLIT_IN_GAG(x1, x2, x3) = SPLIT_IN_GAG(x1, x3)
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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:

PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAG(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → U2_GAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_GAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → U2_AAG(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → U1_GAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_GAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → U1_AAA(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
PERM_IN_AA(XS, cons(Y, YS)) → SPLIT_IN_AAG(XS, YS1, cons(Y, YS2))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → APPEND_IN_GAA(YS1, YS2, ZS)
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_AA(XS, Y, YS, perm_in_aa(ZS, YS))
U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

↳ 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:

APPEND_IN_AAA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_AAA(XS1, XS2, YS)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

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

APPEND_IN_AAA → APPEND_IN_AAA

The TRS R consists of the following rules:none

s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA

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 APPEND_IN_AAA to APPEND_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

↳ 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:

SPLIT_IN_AAG(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_AAG(XS, YS1, YS2)

↳ 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:

SPLIT_IN_AAG(YS2) → SPLIT_IN_AAG(YS2)

The TRS R consists of the following rules:none

s = SPLIT_IN_AAG(YS2) evaluates to t =SPLIT_IN_AAG(YS2)

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 SPLIT_IN_AAG(YS2) to SPLIT_IN_AAG(YS2).

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gag(XS, YS1, cons(Y, YS2)))
split_in_gag(XS, nil, XS) → split_out_gag(XS, nil, XS)
split_in_gag(cons(X, XS), cons(X, YS1), YS2) → U2_gag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
U2_gag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_gag(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gag(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_aa(ZS, YS))
perm_in_aa(nil, nil) → perm_out_aa(nil, nil)
perm_in_aa(XS, cons(Y, YS)) → U3_aa(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))
U3_aa(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_aa(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
U4_aa(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → U5_aa(XS, Y, YS, perm_in_aa(ZS, YS))
U5_aa(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_aa(XS, cons(Y, YS))
U5_ga(XS, Y, YS, perm_out_aa(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
nil = nil
perm_out_ga(x1, x2) = perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4)
split_in_gag(x1, x2, x3) = split_in_gag(x1, x3)
cons(x1, x2) = cons
split_out_gag(x1, x2, x3) = split_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x4, x5)
split_in_aag(x1, x2, x3) = split_in_aag(x3)
split_out_aag(x1, x2, x3) = split_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x1, x6)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
append_out_gaa(x1, x2, x3) = append_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa(x1)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4)
perm_in_aa(x1, x2) = perm_in_aa
perm_out_aa(x1, x2) = perm_out_aa(x1, x2)
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6) = U4_aa(x1, x6)
U5_aa(x1, x2, x3, x4) = U5_aa(x1, x4)
U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x1, x6)
PERM_IN_AA(x1, x2) = PERM_IN_AA
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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:

U4_AA(XS, Y, YS, YS1, YS2, append_out_gaa(YS1, YS2, ZS)) → PERM_IN_AA(ZS, YS)
U3_AA(XS, Y, YS, split_out_aag(XS, YS1, cons(Y, YS2))) → U4_AA(XS, Y, YS, YS1, YS2, append_in_gaa(YS1, YS2, ZS))
PERM_IN_AA(XS, cons(Y, YS)) → U3_AA(XS, Y, YS, split_in_aag(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

append_in_gaa(nil, XS, XS) → append_out_gaa(nil, XS, XS)
append_in_gaa(cons(X, XS1), XS2, cons(X, YS)) → U1_gaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
split_in_aag(XS, nil, XS) → split_out_aag(XS, nil, XS)
split_in_aag(cons(X, XS), cons(X, YS1), YS2) → U2_aag(X, XS, YS1, YS2, split_in_aag(XS, YS1, YS2))
U1_gaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_gaa(cons(X, XS1), XS2, cons(X, YS))
U2_aag(X, XS, YS1, YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons(X, XS), cons(X, YS1), YS2)
append_in_aaa(nil, XS, XS) → append_out_aaa(nil, XS, XS)
append_in_aaa(cons(X, XS1), XS2, cons(X, YS)) → U1_aaa(X, XS1, XS2, YS, append_in_aaa(XS1, XS2, YS))
U1_aaa(X, XS1, XS2, YS, append_out_aaa(XS1, XS2, YS)) → append_out_aaa(cons(X, XS1), XS2, cons(X, YS))

The argument filtering Pi contains the following mapping:
nil = nil
cons(x1, x2) = cons
split_in_aag(x1, x2, x3) = split_in_aag(x3)
split_out_aag(x1, x2, x3) = split_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
append_out_gaa(x1, x2, x3) = append_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa(x1)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x1, x6)
PERM_IN_AA(x1, x2) = PERM_IN_AA
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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:

U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA
PERM_IN_AA → U3_AA(split_in_aag(cons))
U3_AA(split_out_aag(XS, YS1, cons)) → U4_AA(XS, append_in_gaa(YS1))

The TRS R consists of the following rules:

append_in_gaa(nil) → append_out_gaa(nil)
append_in_gaa(cons) → U1_gaa(append_in_aaa)
split_in_aag(XS) → split_out_aag(XS, nil, XS)
split_in_aag(YS2) → U2_aag(YS2, split_in_aag(YS2))
U1_gaa(append_out_aaa(XS1)) → append_out_gaa(cons)
U2_aag(YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons, cons, YS2)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

append_in_gaa(x0)
split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0, x1)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U3_AA(split_out_aag(XS, YS1, cons)) → U4_AA(XS, append_in_gaa(YS1)) at position [1] we obtained the following new rules:

U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil))
U3_AA(split_out_aag(y0, cons, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

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

U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA
U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil))
PERM_IN_AA → U3_AA(split_in_aag(cons))
U3_AA(split_out_aag(y0, cons, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))

The TRS R consists of the following rules:

append_in_gaa(nil) → append_out_gaa(nil)
append_in_gaa(cons) → U1_gaa(append_in_aaa)
split_in_aag(XS) → split_out_aag(XS, nil, XS)
split_in_aag(YS2) → U2_aag(YS2, split_in_aag(YS2))
U1_gaa(append_out_aaa(XS1)) → append_out_gaa(cons)
U2_aag(YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons, cons, YS2)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

append_in_gaa(x0)
split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0, x1)
append_in_aaa
U1_aaa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

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

U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA
U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil))
PERM_IN_AA → U3_AA(split_in_aag(cons))
U3_AA(split_out_aag(y0, cons, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil, XS)
split_in_aag(YS2) → U2_aag(YS2, split_in_aag(YS2))
U2_aag(YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons, cons, YS2)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa(cons)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

append_in_gaa(x0)
split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0, x1)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

append_in_gaa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

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

U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA
PERM_IN_AA → U3_AA(split_in_aag(cons))
U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil))
U3_AA(split_out_aag(y0, cons, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil, XS)
split_in_aag(YS2) → U2_aag(YS2, split_in_aag(YS2))
U2_aag(YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons, cons, YS2)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa(cons)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0, x1)
append_in_aaa
U1_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PERM_IN_AA → U3_AA(split_in_aag(cons)) at position [0] we obtained the following new rules:

PERM_IN_AA → U3_AA(split_out_aag(cons, nil, cons))
PERM_IN_AA → U3_AA(U2_aag(cons, split_in_aag(cons)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ NonTerminationProof

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

U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA
PERM_IN_AA → U3_AA(U2_aag(cons, split_in_aag(cons)))
PERM_IN_AA → U3_AA(split_out_aag(cons, nil, cons))
U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil))
U3_AA(split_out_aag(y0, cons, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil, XS)
split_in_aag(YS2) → U2_aag(YS2, split_in_aag(YS2))
U2_aag(YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons, cons, YS2)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa(cons)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

The set Q consists of the following terms:

split_in_aag(x0)
U1_gaa(x0)
U2_aag(x0, x1)
append_in_aaa
U1_aaa(x0)

U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA
PERM_IN_AA → U3_AA(U2_aag(cons, split_in_aag(cons)))
PERM_IN_AA → U3_AA(split_out_aag(cons, nil, cons))
U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil))
U3_AA(split_out_aag(y0, cons, cons)) → U4_AA(y0, U1_gaa(append_in_aaa))

The TRS R consists of the following rules:

split_in_aag(XS) → split_out_aag(XS, nil, XS)
split_in_aag(YS2) → U2_aag(YS2, split_in_aag(YS2))
U2_aag(YS2, split_out_aag(XS, YS1, YS2)) → split_out_aag(cons, cons, YS2)
append_in_aaa → append_out_aaa(nil)
append_in_aaa → U1_aaa(append_in_aaa)
U1_gaa(append_out_aaa(XS1)) → append_out_gaa(cons)
U1_aaa(append_out_aaa(XS1)) → append_out_aaa(cons)

s = PERM_IN_AA evaluates to t =PERM_IN_AA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

PERM_IN_AA → U3_AA(split_out_aag(cons, nil, cons))
with rule PERM_IN_AA → U3_AA(split_out_aag(cons, nil, cons)) at position [] and matcher [ ]

U3_AA(split_out_aag(cons, nil, cons)) → U4_AA(cons, append_out_gaa(nil))
with rule U3_AA(split_out_aag(y0, nil, cons)) → U4_AA(y0, append_out_gaa(nil)) at position [] and matcher [y0 / cons]

U4_AA(cons, append_out_gaa(nil)) → PERM_IN_AA
with rule U4_AA(XS, append_out_gaa(YS1)) → PERM_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.