(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), list(Xs)).
list([]).
list(X) :- ','(no(empty(X)), ','(tail(X, T), list(T))).
s2l(0, []).
s2l(X, .(X1, Xs)) :- ','(no(zero(X)), ','(p(X, P), s2l(P, Xs))).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
empty([]).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X3).
failure(b).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

s2l4(0, []).
s2l4(s(T7), .(X14, X15)) :- s2l4(T7, X15).
list5([]).
list5([]).
list5(.(T14, T16)) :- list5(T16).
list5(.(T17, T19)) :- list5(T19).
goal1(T2) :- s2l4(T2, X6).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- ','(s2l4(T2, .(T14, T16)), list5(T16)).
goal1(T2) :- ','(s2l4(T2, .(T17, T19)), list5(T19)).

Queries:

goal1(g).

(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:
goal1_in: (b)
s2l4_in: (b,f) (b,b)
list5_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

(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:

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X6))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X6)
S2L4_IN_GA(s(T7), .(X14, X15)) → U1_GA(T7, X14, X15, s2l4_in_ga(T7, X15))
S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T7), .(X14, X15)) → U1_GG(T7, X14, X15, s2l4_in_gg(T7, X15))
S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T14, T16)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T14, T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_G(T2, list5_in_g(T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → LIST5_IN_G(T16)
LIST5_IN_G(.(T14, T16)) → U2_G(T14, T16, list5_in_g(T16))
LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)
LIST5_IN_G(.(T17, T19)) → U3_G(T17, T19, list5_in_g(T19))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_G(T2, list5_in_g(T19))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → LIST5_IN_G(T19)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)
U3_G(x1, x2, x3)  =  U3_G(x3)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(6) Obligation:

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

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X6))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X6)
S2L4_IN_GA(s(T7), .(X14, X15)) → U1_GA(T7, X14, X15, s2l4_in_ga(T7, X15))
S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T7), .(X14, X15)) → U1_GG(T7, X14, X15, s2l4_in_gg(T7, X15))
S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T14, T16)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T14, T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_G(T2, list5_in_g(T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → LIST5_IN_G(T16)
LIST5_IN_G(.(T14, T16)) → U2_G(T14, T16, list5_in_g(T16))
LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)
LIST5_IN_G(.(T17, T19)) → U3_G(T17, T19, list5_in_g(T19))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_G(T2, list5_in_g(T19))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → LIST5_IN_G(T19)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)
U3_G(x1, x2, x3)  =  U3_G(x3)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)

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:

LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)

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:

LIST5_IN_G(.(T16)) → LIST5_IN_G(T16)

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:

  • LIST5_IN_G(.(T16)) → LIST5_IN_G(T16)
    The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T7), .(X15)) → S2L4_IN_GG(T7, X15)

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:

  • S2L4_IN_GG(s(T7), .(X15)) → S2L4_IN_GG(T7, X15)
    The graph contains the following edges 1 > 1, 2 > 2

(22) TRUE

(23) Obligation:

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

S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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:

S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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:

S2L4_IN_GA(s(T7)) → S2L4_IN_GA(T7)

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

(28) 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:

  • S2L4_IN_GA(s(T7)) → S2L4_IN_GA(T7)
    The graph contains the following edges 1 > 1

(29) TRUE