(0) Obligation:

Clauses:

append2(parts([], Y), is(sum(Y))).
append2(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append2(parts(X, Y), is(sum(Z))).
append1(parts([], Y), is(sum(Y))).
append1(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append1(parts(X, Y), is(sum(Z))).
perm([], []).
perm(L, .(H, T)) :- ','(append2(parts(V, .(H, U)), is(sum(L))), ','(append1(parts(V, U), is(sum(W))), perm(W, T))).

Queries:

perm(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

append234([], T50, X98, .(T50, X98)).
append234(.(X118, X119), T57, X120, .(X118, T58)) :- append234(X119, T57, X120, T58).
perm19([], []).
perm19(T37, .(T36, T38)) :- append234(X75, T36, X76, T37).
perm19(T37, .(T36, T43)) :- ','(append234(T41, T36, T42, T37), p35(T41, T42, X77, T43)).
append144([], T71, T71).
append144(.(T78, T81), T82, .(T78, X153)) :- append144(T81, T82, X153).
p35(T41, T42, X77, T43) :- append144(T41, T42, X77).
p35(T41, T42, T63, T64) :- ','(append144(T41, T42, T63), perm19(T63, T64)).
append118(X58, X58).
perm1([], []).
perm1(.(T23, X43), .(T23, T24)) :- append118(X43, X30).
perm1(.(T23, T25), .(T23, T24)) :- ','(append118(T25, T26), perm19(T26, T24)).
perm1(.(X178, T90), .(T91, T92)) :- append234(X179, T91, X180, T90).
perm1(.(X178, T90), .(T91, T97)) :- ','(append234(T95, T91, T96, T90), p35(.(X178, T95), T96, X30, T97)).

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)
perm19_in: (b,f)
append234_in: (f,f,f,b)
p35_in: (b,b,f,f)
append144_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([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_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([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_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(.(T23, X43), .(T23, T24)) → U9_GA(T23, X43, T24, append118_in_ga(X43, X30))
PERM1_IN_GA(.(T23, X43), .(T23, T24)) → APPEND118_IN_GA(X43, X30)
PERM1_IN_GA(.(T23, T25), .(T23, T24)) → U10_GA(T23, T25, T24, append118_in_ga(T25, T26))
U10_GA(T23, T25, T24, append118_out_ga(T25, T26)) → U11_GA(T23, T25, T24, perm19_in_ga(T26, T24))
U10_GA(T23, T25, T24, append118_out_ga(T25, T26)) → PERM19_IN_GA(T26, T24)
PERM19_IN_GA(T37, .(T36, T38)) → U2_GA(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
PERM19_IN_GA(T37, .(T36, T38)) → APPEND234_IN_AAAG(X75, T36, X76, T37)
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → U1_AAAG(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)
PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_GA(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
U3_GA(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, X77, T43) → U6_GGAA(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
P35_IN_GGAA(T41, T42, X77, T43) → APPEND144_IN_GGA(T41, T42, X77)
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → U5_GGA(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_GGAA(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U7_GGAA(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → U12_GA(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → APPEND234_IN_AAAG(X179, T91, X180, T90)
PERM1_IN_GA(.(X178, T90), .(T91, T97)) → U13_GA(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_GA(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_GA(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U13_GA(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → P35_IN_GGAA(.(X178, T95), T96, X30, T97)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
APPEND118_IN_GA(x1, x2)  =  APPEND118_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5)  =  U6_GGAA(x5)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x5)
U8_GGAA(x1, x2, x3, x4, x5)  =  U8_GGAA(x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_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(.(T23, X43), .(T23, T24)) → U9_GA(T23, X43, T24, append118_in_ga(X43, X30))
PERM1_IN_GA(.(T23, X43), .(T23, T24)) → APPEND118_IN_GA(X43, X30)
PERM1_IN_GA(.(T23, T25), .(T23, T24)) → U10_GA(T23, T25, T24, append118_in_ga(T25, T26))
U10_GA(T23, T25, T24, append118_out_ga(T25, T26)) → U11_GA(T23, T25, T24, perm19_in_ga(T26, T24))
U10_GA(T23, T25, T24, append118_out_ga(T25, T26)) → PERM19_IN_GA(T26, T24)
PERM19_IN_GA(T37, .(T36, T38)) → U2_GA(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
PERM19_IN_GA(T37, .(T36, T38)) → APPEND234_IN_AAAG(X75, T36, X76, T37)
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → U1_AAAG(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)
PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_GA(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
U3_GA(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, X77, T43) → U6_GGAA(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
P35_IN_GGAA(T41, T42, X77, T43) → APPEND144_IN_GGA(T41, T42, X77)
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → U5_GGA(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_GGAA(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U7_GGAA(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → U12_GA(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → APPEND234_IN_AAAG(X179, T91, X180, T90)
PERM1_IN_GA(.(X178, T90), .(T91, T97)) → U13_GA(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_GA(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_GA(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U13_GA(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → P35_IN_GGAA(.(X178, T95), T96, X30, T97)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
APPEND118_IN_GA(x1, x2)  =  APPEND118_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5)  =  U6_GGAA(x5)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x5)
U8_GGAA(x1, x2, x3, x4, x5)  =  U8_GGAA(x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_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 18 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_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:

APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_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:

APPEND144_IN_GGA(.(T78, T81), T82) → APPEND144_IN_GGA(T81, T82)

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:

  • APPEND144_IN_GGA(.(T78, T81), T82) → APPEND144_IN_GGA(T81, T82)
    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:

APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)

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:

APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)

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:

APPEND234_IN_AAAG(.(X118, T58)) → APPEND234_IN_AAAG(T58)

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:

  • APPEND234_IN_AAAG(.(X118, T58)) → APPEND234_IN_AAAG(T58)
    The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T23, X43), .(T23, T24)) → U9_ga(T23, X43, T24, append118_in_ga(X43, X30))
append118_in_ga(X58, X58) → append118_out_ga(X58, X58)
U9_ga(T23, X43, T24, append118_out_ga(X43, X30)) → perm1_out_ga(.(T23, X43), .(T23, T24))
perm1_in_ga(.(T23, T25), .(T23, T24)) → U10_ga(T23, T25, T24, append118_in_ga(T25, T26))
U10_ga(T23, T25, T24, append118_out_ga(T25, T26)) → U11_ga(T23, T25, T24, perm19_in_ga(T26, T24))
perm19_in_ga([], []) → perm19_out_ga([], [])
perm19_in_ga(T37, .(T36, T38)) → U2_ga(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U2_ga(T37, T36, T38, append234_out_aaag(X75, T36, X76, T37)) → perm19_out_ga(T37, .(T36, T38))
perm19_in_ga(T37, .(T36, T43)) → U3_ga(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_ga(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → U4_ga(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
p35_in_ggaa(T41, T42, X77, T43) → U6_ggaa(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))
U6_ggaa(T41, T42, X77, T43, append144_out_gga(T41, T42, X77)) → p35_out_ggaa(T41, T42, X77, T43)
p35_in_ggaa(T41, T42, T63, T64) → U7_ggaa(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_ggaa(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → U8_ggaa(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U8_ggaa(T41, T42, T63, T64, perm19_out_ga(T63, T64)) → p35_out_ggaa(T41, T42, T63, T64)
U4_ga(T37, T36, T43, p35_out_ggaa(T41, T42, X77, T43)) → perm19_out_ga(T37, .(T36, T43))
U11_ga(T23, T25, T24, perm19_out_ga(T26, T24)) → perm1_out_ga(.(T23, T25), .(T23, T24))
perm1_in_ga(.(X178, T90), .(T91, T92)) → U12_ga(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
U12_ga(X178, T90, T91, T92, append234_out_aaag(X179, T91, X180, T90)) → perm1_out_ga(.(X178, T90), .(T91, T92))
perm1_in_ga(.(X178, T90), .(T91, T97)) → U13_ga(X178, T90, T91, T97, append234_in_aaag(T95, T91, T96, T90))
U13_ga(X178, T90, T91, T97, append234_out_aaag(T95, T91, T96, T90)) → U14_ga(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U14_ga(X178, T90, T91, T97, p35_out_ggaa(.(X178, T95), T96, X30, T97)) → perm1_out_ga(.(X178, T90), .(T91, T97))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga
.(x1, x2)  =  .(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
append118_in_ga(x1, x2)  =  append118_in_ga(x1)
append118_out_ga(x1, x2)  =  append118_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
perm19_out_ga(x1, x2)  =  perm19_out_ga
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5)  =  U6_ggaa(x5)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
p35_out_ggaa(x1, x2, x3, x4)  =  p35_out_ggaa(x3)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(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:

PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append234_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append234_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append144_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append144_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)

The TRS R consists of the following rules:

append234_in_aaag([], T50, X98, .(T50, X98)) → append234_out_aaag([], T50, X98, .(T50, X98))
append234_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U1_aaag(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
append144_in_gga([], T71, T71) → append144_out_gga([], T71, T71)
append144_in_gga(.(T78, T81), T82, .(T78, X153)) → U5_gga(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
U1_aaag(X118, X119, T57, X120, T58, append234_out_aaag(X119, T57, X120, T58)) → append234_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U5_gga(T78, T81, T82, X153, append144_out_gga(T81, T82, X153)) → append144_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append234_out_aaag(x1, x2, x3, x4)  =  append234_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append144_out_gga(x1, x2, x3)  =  append144_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(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:

PERM19_IN_GA(T37) → U3_GA(append234_in_aaag(T37))
U3_GA(append234_out_aaag(T41, T36, T42)) → P35_IN_GGAA(T41, T42)
P35_IN_GGAA(T41, T42) → U7_GGAA(append144_in_gga(T41, T42))
U7_GGAA(append144_out_gga(T63)) → PERM19_IN_GA(T63)

The TRS R consists of the following rules:

append234_in_aaag(.(T50, X98)) → append234_out_aaag([], T50, X98)
append234_in_aaag(.(X118, T58)) → U1_aaag(X118, append234_in_aaag(T58))
append144_in_gga([], T71) → append144_out_gga(T71)
append144_in_gga(.(T78, T81), T82) → U5_gga(T78, append144_in_gga(T81, T82))
U1_aaag(X118, append234_out_aaag(X119, T57, X120)) → append234_out_aaag(.(X118, X119), T57, X120)
U5_gga(T78, append144_out_gga(X153)) → append144_out_gga(.(T78, X153))

The set Q consists of the following terms:

append234_in_aaag(x0)
append144_in_gga(x0, x1)
U1_aaag(x0, x1)
U5_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:

PERM19_IN_GA(T37) → U3_GA(append234_in_aaag(T37))
U3_GA(append234_out_aaag(T41, T36, T42)) → P35_IN_GGAA(T41, T42)
P35_IN_GGAA(T41, T42) → U7_GGAA(append144_in_gga(T41, T42))
U7_GGAA(append144_out_gga(T63)) → PERM19_IN_GA(T63)

Strictly oriented rules of the TRS R:

append234_in_aaag(.(T50, X98)) → append234_out_aaag([], T50, X98)
append144_in_gga([], T71) → append144_out_gga(T71)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 6 + x1 + x2   
POL(P35_IN_GGAA(x1, x2)) = 4 + x1 + x2   
POL(PERM19_IN_GA(x1)) = 1 + x1   
POL(U1_aaag(x1, x2)) = 6 + x1 + x2   
POL(U3_GA(x1)) = x1   
POL(U5_gga(x1, x2)) = 6 + x1 + x2   
POL(U7_GGAA(x1)) = x1   
POL([]) = 0   
POL(append144_in_gga(x1, x2)) = 3 + x1 + x2   
POL(append144_out_gga(x1)) = 2 + x1   
POL(append234_in_aaag(x1)) = x1   
POL(append234_out_aaag(x1, x2, x3)) = 5 + x1 + x2 + x3   

(29) Obligation:

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

append234_in_aaag(.(X118, T58)) → U1_aaag(X118, append234_in_aaag(T58))
append144_in_gga(.(T78, T81), T82) → U5_gga(T78, append144_in_gga(T81, T82))
U1_aaag(X118, append234_out_aaag(X119, T57, X120)) → append234_out_aaag(.(X118, X119), T57, X120)
U5_gga(T78, append144_out_gga(X153)) → append144_out_gga(.(T78, X153))

The set Q consists of the following terms:

append234_in_aaag(x0)
append144_in_gga(x0, x1)
U1_aaag(x0, x1)
U5_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