Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇐)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔)
        ↳22 YES
    ↳23 PiDP
        ↳24 UsableRulesProof (⇔)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇐)
        ↳27 QDP
        ↳28 MRRProof (⇔)
        ↳29 QDP
        ↳30 PisEmptyProof (⇔)
        ↳31 YES

(0) Obligation:

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).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

split28(cons(T80, T81), nil, T80, T81).
split28(cons(T88, T89), cons(T88, X95), T91, X96) :- split28(T89, X95, T91, X96).
append42(nil, T122, T122).
append42(cons(T129, T130), T131, cons(T129, X157)) :- append42(T130, T131, X157).
append18(T40, T40).
append38(T113, T114, T115, cons(T113, X134)) :- append42(T114, T115, X134).
perm1(nil, nil).
perm1(cons(T30, T31), cons(T30, T32)) :- append18(T31, X26).
perm1(cons(T30, T31), cons(T30, T32)) :- ','(append18(T31, T34), perm1(T34, T32)).
perm1(cons(T47, T48), cons(T50, T51)) :- split28(T48, X63, T50, X64).
perm1(cons(T47, T48), cons(T50, T60)) :- ','(split28(T48, T58, T50, T59), append38(T47, T58, T59, X26)).
perm1(cons(T47, T48), cons(T50, T60)) :- ','(split28(T48, T58, T50, T59), ','(append38(T47, T58, T59, T101), perm1(T101, T60))).

Queries:

perm1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm1_in: (b,f)
split28_in: (b,f,f,f)
append38_in: (b,b,b,f)
append42_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → U4_GA(T30, T31, T32, append18_in_ga(T31, X26))
PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → APPEND18_IN_GA(T31, X26)
PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, append18_in_ga(T31, T34))
U5_GA(T30, T31, T32, append18_out_ga(T31, T34)) → U6_GA(T30, T31, T32, perm1_in_ga(T34, T32))
U5_GA(T30, T31, T32, append18_out_ga(T31, T34)) → PERM1_IN_GA(T34, T32)
PERM1_IN_GA(cons(T47, T48), cons(T50, T51)) → U7_GA(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
PERM1_IN_GA(cons(T47, T48), cons(T50, T51)) → SPLIT28_IN_GAAA(T48, X63, T50, X64)
SPLIT28_IN_GAAA(cons(T88, T89), cons(T88, X95), T91, X96) → U1_GAAA(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
SPLIT28_IN_GAAA(cons(T88, T89), cons(T88, X95), T91, X96) → SPLIT28_IN_GAAA(T89, X95, T91, X96)
PERM1_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_GA(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → APPEND38_IN_GGGA(T47, T58, T59, X26)
APPEND38_IN_GGGA(T113, T114, T115, cons(T113, X134)) → U3_GGGA(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
APPEND38_IN_GGGA(T113, T114, T115, cons(T113, X134)) → APPEND42_IN_GGA(T114, T115, X134)
APPEND42_IN_GGA(cons(T129, T130), T131, cons(T129, X157)) → U2_GGA(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
APPEND42_IN_GGA(cons(T129, T130), T131, cons(T129, X157)) → APPEND42_IN_GGA(T130, T131, X157)
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_GA(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_GA(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U10_GA(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → PERM1_IN_GA(T101, T60)

The TRS R consists of the following rules:

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APPEND18_IN_GA(x1, x2)  =  APPEND18_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
SPLIT28_IN_GAAA(x1, x2, x3, x4)  =  SPLIT28_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
APPEND38_IN_GGGA(x1, x2, x3, x4)  =  APPEND38_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x5)
APPEND42_IN_GGA(x1, x2, x3)  =  APPEND42_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)

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

(6) Obligation:

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

PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → U4_GA(T30, T31, T32, append18_in_ga(T31, X26))
PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → APPEND18_IN_GA(T31, X26)
PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, append18_in_ga(T31, T34))
U5_GA(T30, T31, T32, append18_out_ga(T31, T34)) → U6_GA(T30, T31, T32, perm1_in_ga(T34, T32))
U5_GA(T30, T31, T32, append18_out_ga(T31, T34)) → PERM1_IN_GA(T34, T32)
PERM1_IN_GA(cons(T47, T48), cons(T50, T51)) → U7_GA(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
PERM1_IN_GA(cons(T47, T48), cons(T50, T51)) → SPLIT28_IN_GAAA(T48, X63, T50, X64)
SPLIT28_IN_GAAA(cons(T88, T89), cons(T88, X95), T91, X96) → U1_GAAA(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
SPLIT28_IN_GAAA(cons(T88, T89), cons(T88, X95), T91, X96) → SPLIT28_IN_GAAA(T89, X95, T91, X96)
PERM1_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_GA(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → APPEND38_IN_GGGA(T47, T58, T59, X26)
APPEND38_IN_GGGA(T113, T114, T115, cons(T113, X134)) → U3_GGGA(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
APPEND38_IN_GGGA(T113, T114, T115, cons(T113, X134)) → APPEND42_IN_GGA(T114, T115, X134)
APPEND42_IN_GGA(cons(T129, T130), T131, cons(T129, X157)) → U2_GGA(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
APPEND42_IN_GGA(cons(T129, T130), T131, cons(T129, X157)) → APPEND42_IN_GGA(T130, T131, X157)
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_GA(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_GA(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U10_GA(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → PERM1_IN_GA(T101, T60)

The TRS R consists of the following rules:

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APPEND18_IN_GA(x1, x2)  =  APPEND18_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
SPLIT28_IN_GAAA(x1, x2, x3, x4)  =  SPLIT28_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
APPEND38_IN_GGGA(x1, x2, x3, x4)  =  APPEND38_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x5)
APPEND42_IN_GGA(x1, x2, x3)  =  APPEND42_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND42_IN_GGA(cons(T129, T130), T131, cons(T129, X157)) → APPEND42_IN_GGA(T130, T131, X157)

The TRS R consists of the following rules:

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
APPEND42_IN_GGA(x1, x2, x3)  =  APPEND42_IN_GGA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPEND42_IN_GGA(cons(T129, T130), T131, cons(T129, X157)) → APPEND42_IN_GGA(T130, T131, X157)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND42_IN_GGA(x1, x2, x3)  =  APPEND42_IN_GGA(x1, x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APPEND42_IN_GGA(cons(T129, T130), T131) → APPEND42_IN_GGA(T130, T131)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • APPEND42_IN_GGA(cons(T129, T130), T131) → APPEND42_IN_GGA(T130, T131)
    The graph contains the following edges 1 > 1, 2 >= 2

(15) YES

(16) Obligation:

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

SPLIT28_IN_GAAA(cons(T88, T89), cons(T88, X95), T91, X96) → SPLIT28_IN_GAAA(T89, X95, T91, X96)

The TRS R consists of the following rules:

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
SPLIT28_IN_GAAA(x1, x2, x3, x4)  =  SPLIT28_IN_GAAA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

SPLIT28_IN_GAAA(cons(T88, T89), cons(T88, X95), T91, X96) → SPLIT28_IN_GAAA(T89, X95, T91, X96)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
SPLIT28_IN_GAAA(x1, x2, x3, x4)  =  SPLIT28_IN_GAAA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

SPLIT28_IN_GAAA(cons(T88, T89)) → SPLIT28_IN_GAAA(T89)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • SPLIT28_IN_GAAA(cons(T88, T89)) → SPLIT28_IN_GAAA(T89)
    The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, append18_in_ga(T31, T34))
U5_GA(T30, T31, T32, append18_out_ga(T31, T34)) → PERM1_IN_GA(T34, T32)
PERM1_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_GA(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → PERM1_IN_GA(T101, T60)

The TRS R consists of the following rules:

perm1_in_ga(nil, nil) → perm1_out_ga(nil, nil)
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, append18_in_ga(T31, X26))
append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
U4_ga(T30, T31, T32, append18_out_ga(T31, X26)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))
perm1_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, append18_in_ga(T31, T34))
U5_ga(T30, T31, T32, append18_out_ga(T31, T34)) → U6_ga(T30, T31, T32, perm1_in_ga(T34, T32))
perm1_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, split28_in_gaaa(T48, X63, T50, X64))
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U7_ga(T47, T48, T50, T51, split28_out_gaaa(T48, X63, T50, X64)) → perm1_out_ga(cons(T47, T48), cons(T50, T51))
perm1_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, X26))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
U9_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, X26)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_ga(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → U11_ga(T47, T48, T50, T60, perm1_in_ga(T101, T60))
U11_ga(T47, T48, T50, T60, perm1_out_ga(T101, T60)) → perm1_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, perm1_out_ga(T34, T32)) → perm1_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
nil  =  nil
perm1_out_ga(x1, x2)  =  perm1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

PERM1_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, append18_in_ga(T31, T34))
U5_GA(T30, T31, T32, append18_out_ga(T31, T34)) → PERM1_IN_GA(T34, T32)
PERM1_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, split28_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, split28_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, append38_in_ggga(T47, T58, T59, T101))
U10_GA(T47, T48, T50, T60, append38_out_ggga(T47, T58, T59, T101)) → PERM1_IN_GA(T101, T60)

The TRS R consists of the following rules:

append18_in_ga(T40, T40) → append18_out_ga(T40, T40)
split28_in_gaaa(cons(T80, T81), nil, T80, T81) → split28_out_gaaa(cons(T80, T81), nil, T80, T81)
split28_in_gaaa(cons(T88, T89), cons(T88, X95), T91, X96) → U1_gaaa(T88, T89, X95, T91, X96, split28_in_gaaa(T89, X95, T91, X96))
append38_in_ggga(T113, T114, T115, cons(T113, X134)) → U3_ggga(T113, T114, T115, X134, append42_in_gga(T114, T115, X134))
U1_gaaa(T88, T89, X95, T91, X96, split28_out_gaaa(T89, X95, T91, X96)) → split28_out_gaaa(cons(T88, T89), cons(T88, X95), T91, X96)
U3_ggga(T113, T114, T115, X134, append42_out_gga(T114, T115, X134)) → append38_out_ggga(T113, T114, T115, cons(T113, X134))
append42_in_gga(nil, T122, T122) → append42_out_gga(nil, T122, T122)
append42_in_gga(cons(T129, T130), T131, cons(T129, X157)) → U2_gga(T129, T130, T131, X157, append42_in_gga(T130, T131, X157))
U2_gga(T129, T130, T131, X157, append42_out_gga(T130, T131, X157)) → append42_out_gga(cons(T129, T130), T131, cons(T129, X157))

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
append18_in_ga(x1, x2)  =  append18_in_ga(x1)
append18_out_ga(x1, x2)  =  append18_out_ga(x2)
split28_in_gaaa(x1, x2, x3, x4)  =  split28_in_gaaa(x1)
split28_out_gaaa(x1, x2, x3, x4)  =  split28_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
append38_in_ggga(x1, x2, x3, x4)  =  append38_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
append42_in_gga(x1, x2, x3)  =  append42_in_gga(x1, x2)
append42_out_gga(x1, x2, x3)  =  append42_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
append38_out_ggga(x1, x2, x3, x4)  =  append38_out_ggga(x4)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

PERM1_IN_GA(cons(T30, T31)) → U5_GA(append18_in_ga(T31))
U5_GA(append18_out_ga(T34)) → PERM1_IN_GA(T34)
PERM1_IN_GA(cons(T47, T48)) → U8_GA(T47, split28_in_gaaa(T48))
U8_GA(T47, split28_out_gaaa(T58, T50, T59)) → U10_GA(append38_in_ggga(T47, T58, T59))
U10_GA(append38_out_ggga(T101)) → PERM1_IN_GA(T101)

The TRS R consists of the following rules:

append18_in_ga(T40) → append18_out_ga(T40)
split28_in_gaaa(cons(T80, T81)) → split28_out_gaaa(nil, T80, T81)
split28_in_gaaa(cons(T88, T89)) → U1_gaaa(T88, split28_in_gaaa(T89))
append38_in_ggga(T113, T114, T115) → U3_ggga(T113, append42_in_gga(T114, T115))
U1_gaaa(T88, split28_out_gaaa(X95, T91, X96)) → split28_out_gaaa(cons(T88, X95), T91, X96)
U3_ggga(T113, append42_out_gga(X134)) → append38_out_ggga(cons(T113, X134))
append42_in_gga(nil, T122) → append42_out_gga(T122)
append42_in_gga(cons(T129, T130), T131) → U2_gga(T129, append42_in_gga(T130, T131))
U2_gga(T129, append42_out_gga(X157)) → append42_out_gga(cons(T129, X157))

The set Q consists of the following terms:

append18_in_ga(x0)
split28_in_gaaa(x0)
append38_in_ggga(x0, x1, x2)
U1_gaaa(x0, x1)
U3_ggga(x0, x1)
append42_in_gga(x0, x1)
U2_gga(x0, x1)

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

(28) MRRProof (EQUIVALENT transformation)

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

PERM1_IN_GA(cons(T30, T31)) → U5_GA(append18_in_ga(T31))
U5_GA(append18_out_ga(T34)) → PERM1_IN_GA(T34)
PERM1_IN_GA(cons(T47, T48)) → U8_GA(T47, split28_in_gaaa(T48))
U8_GA(T47, split28_out_gaaa(T58, T50, T59)) → U10_GA(append38_in_ggga(T47, T58, T59))
U10_GA(append38_out_ggga(T101)) → PERM1_IN_GA(T101)

Strictly oriented rules of the TRS R:

append18_in_ga(T40) → append18_out_ga(T40)
split28_in_gaaa(cons(T80, T81)) → split28_out_gaaa(nil, T80, T81)
append38_in_ggga(T113, T114, T115) → U3_ggga(T113, append42_in_gga(T114, T115))
U3_ggga(T113, append42_out_gga(X134)) → append38_out_ggga(cons(T113, X134))
append42_in_gga(nil, T122) → append42_out_gga(T122)

Used ordering: Polynomial interpretation [POLO]:

POL(PERM1_IN_GA(x1)) = x1   
POL(U10_GA(x1)) = x1   
POL(U1_gaaa(x1, x2)) = 7 + x1 + x2   
POL(U2_gga(x1, x2)) = 7 + x1 + x2   
POL(U3_ggga(x1, x2)) = x1 + x2   
POL(U5_GA(x1)) = x1   
POL(U8_GA(x1, x2)) = x1 + x2   
POL(append18_in_ga(x1)) = 2 + x1   
POL(append18_out_ga(x1)) = 1 + x1   
POL(append38_in_ggga(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(append38_out_ggga(x1)) = 1 + x1   
POL(append42_in_gga(x1, x2)) = x1 + x2   
POL(append42_out_gga(x1)) = 9 + x1   
POL(cons(x1, x2)) = 7 + x1 + x2   
POL(nil) = 10   
POL(split28_in_gaaa(x1)) = 6 + x1   
POL(split28_out_gaaa(x1, x2, x3)) = 2 + x1 + x2 + x3   

(29) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

split28_in_gaaa(cons(T88, T89)) → U1_gaaa(T88, split28_in_gaaa(T89))
U1_gaaa(T88, split28_out_gaaa(X95, T91, X96)) → split28_out_gaaa(cons(T88, X95), T91, X96)
append42_in_gga(cons(T129, T130), T131) → U2_gga(T129, append42_in_gga(T130, T131))
U2_gga(T129, append42_out_gga(X157)) → append42_out_gga(cons(T129, X157))

The set Q consists of the following terms:

append18_in_ga(x0)
split28_in_gaaa(x0)
append38_in_ggga(x0, x1, x2)
U1_gaaa(x0, x1)
U3_ggga(x0, x1)
append42_in_gga(x0, x1)
U2_gga(x0, x1)

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

(30) PisEmptyProof (EQUIVALENT transformation)

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

(31) YES