(0) Obligation:

Clauses:

tappend(nil, T, T).
tappend(node(nil, X, T2), T1, node(T1, X, T2)).
tappend(node(T1, X, nil), T2, node(T1, X, T2)).
tappend(node(T1, X, T2), T3, node(U, X, T2)) :- tappend(T1, T3, U).
tappend(node(T1, X, T2), T3, node(T1, X, U)) :- tappend(T2, T3, U).
s2t(s(X), node(T, Y, T)) :- s2t(X, T).
s2t(s(X), node(nil, Y, T)) :- s2t(X, T).
s2t(s(X), node(T, Y, nil)) :- s2t(X, T).
s2t(s(X), node(nil, Y, nil)).
s2t(0, nil).
goal(X) :- ','(s2t(X, T1), tappend(T1, T2, T3)).

Queries:

goal(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

s2t9(s(T19), node(X67, X68, X67)) :- s2t9(T19, X67).
s2t9(s(T25), node(nil, X100, X101)) :- s2t9(T25, X101).
s2t9(s(T31), node(X133, X134, nil)) :- s2t9(T31, X133).
tappend46(node(T86, T84, T85), X371, node(X372, T84, T85)) :- tappend46(T86, X371, X372).
tappend46(node(T93, T94, T96), X401, node(T93, T94, X402)) :- tappend46(T96, X401, X402).
tappend10(T50, X274, X275, node(X276, X274, T50)) :- tappend46(T50, X275, X276).
tappend10(T101, X437, X438, node(T101, X437, X439)) :- tappend46(T101, X438, X439).
goal1(s(T11)) :- s2t9(T11, X29).
goal1(s(T11)) :- ','(s2tc9(T11, T13), tappend10(T13, X30, X4, X5)).
goal1(s(T106)) :- s2t9(T106, X468).
goal1(s(T106)) :- ','(s2tc9(T106, T108), tappend46(node(nil, X467, T108), X4, X5)).
goal1(s(T115)) :- s2t9(T115, X512).
goal1(s(T115)) :- ','(s2tc9(T115, T117), tappend46(node(T117, X513, nil), X4, X5)).
goal1(s(T123)) :- tappend10(nil, X553, X4, X5).
goal1(0) :- tappend46(nil, X4, X5).

Clauses:

s2tc9(s(T19), node(X67, X68, X67)) :- s2tc9(T19, X67).
s2tc9(s(T25), node(nil, X100, X101)) :- s2tc9(T25, X101).
s2tc9(s(T31), node(X133, X134, nil)) :- s2tc9(T31, X133).
s2tc9(s(T37), node(nil, X157, nil)).
s2tc9(0, nil).
tappendc46(nil, X290, X290).
tappendc46(node(nil, T59, T60), X311, node(X311, T59, T60)).
tappendc46(node(T69, T70, nil), X334, node(T69, T70, X334)).
tappendc46(node(T86, T84, T85), X371, node(X372, T84, T85)) :- tappendc46(T86, X371, X372).
tappendc46(node(T93, T94, T96), X401, node(T93, T94, X402)) :- tappendc46(T96, X401, X402).
tappendc10(nil, X196, X197, node(X197, X196, nil)).
tappendc10(nil, X224, X225, node(nil, X224, X225)).
tappendc10(T50, X274, X275, node(X276, X274, T50)) :- tappendc46(T50, X275, X276).
tappendc10(T101, X437, X438, node(T101, X437, X439)) :- tappendc46(T101, X438, X439).

Afs:

goal1(x1)  =  goal1(x1)

(3) TriplesToPiDPProof (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)
s2t9_in: (b,f)
s2tc9_in: (b,f)
tappend10_in: (b,f,f,f)
tappend46_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T11)) → U8_G(T11, s2t9_in_ga(T11, X29))
GOAL1_IN_G(s(T11)) → S2T9_IN_GA(T11, X29)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → U1_GA(T19, X67, X68, s2t9_in_ga(T19, X67))
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → U2_GA(T25, X100, X101, s2t9_in_ga(T25, X101))
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → U3_GA(T31, X133, X134, s2t9_in_ga(T31, X133))
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)
GOAL1_IN_G(s(T11)) → U9_G(T11, s2tc9_in_ga(T11, T13))
U9_G(T11, s2tc9_out_ga(T11, T13)) → U10_G(T11, tappend10_in_gaaa(T13, X30, X4, X5))
U9_G(T11, s2tc9_out_ga(T11, T13)) → TAPPEND10_IN_GAAA(T13, X30, X4, X5)
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → U6_GAAA(T50, X274, X275, X276, tappend46_in_gaa(T50, X275, X276))
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → TAPPEND46_IN_GAA(T50, X275, X276)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → U4_GAA(T86, T84, T85, X371, X372, tappend46_in_gaa(T86, X371, X372))
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → U5_GAA(T93, T94, T96, X401, X402, tappend46_in_gaa(T96, X401, X402))
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → U7_GAAA(T101, X437, X438, X439, tappend46_in_gaa(T101, X438, X439))
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → TAPPEND46_IN_GAA(T101, X438, X439)
GOAL1_IN_G(s(T106)) → U11_G(T106, s2t9_in_ga(T106, X468))
GOAL1_IN_G(s(T106)) → U12_G(T106, s2tc9_in_ga(T106, T108))
U12_G(T106, s2tc9_out_ga(T106, T108)) → U13_G(T106, tappend46_in_gaa(node(nil, X467, T108), X4, X5))
U12_G(T106, s2tc9_out_ga(T106, T108)) → TAPPEND46_IN_GAA(node(nil, X467, T108), X4, X5)
GOAL1_IN_G(s(T115)) → U14_G(T115, s2t9_in_ga(T115, X512))
GOAL1_IN_G(s(T115)) → U15_G(T115, s2tc9_in_ga(T115, T117))
U15_G(T115, s2tc9_out_ga(T115, T117)) → U16_G(T115, tappend46_in_gaa(node(T117, X513, nil), X4, X5))
U15_G(T115, s2tc9_out_ga(T115, T117)) → TAPPEND46_IN_GAA(node(T117, X513, nil), X4, X5)
GOAL1_IN_G(s(T123)) → U17_G(T123, tappend10_in_gaaa(nil, X553, X4, X5))
GOAL1_IN_G(s(T123)) → TAPPEND10_IN_GAAA(nil, X553, X4, X5)
GOAL1_IN_G(0) → U18_G(tappend46_in_gaa(nil, X4, X5))
GOAL1_IN_G(0) → TAPPEND46_IN_GAA(nil, X4, X5)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2t9_in_ga(x1, x2)  =  s2t9_in_ga(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
tappend10_in_gaaa(x1, x2, x3, x4)  =  tappend10_in_gaaa(x1)
tappend46_in_gaa(x1, x2, x3)  =  tappend46_in_gaa(x1)
nil  =  nil
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x1, x2)
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)
TAPPEND10_IN_GAAA(x1, x2, x3, x4)  =  TAPPEND10_IN_GAAA(x1)
U6_GAAA(x1, x2, x3, x4, x5)  =  U6_GAAA(x1, x5)
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x1, x3, x6)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, x3, x6)
U7_GAAA(x1, x2, x3, x4, x5)  =  U7_GAAA(x1, x5)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1)  =  U18_G(x1)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

GOAL1_IN_G(s(T11)) → U8_G(T11, s2t9_in_ga(T11, X29))
GOAL1_IN_G(s(T11)) → S2T9_IN_GA(T11, X29)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → U1_GA(T19, X67, X68, s2t9_in_ga(T19, X67))
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → U2_GA(T25, X100, X101, s2t9_in_ga(T25, X101))
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → U3_GA(T31, X133, X134, s2t9_in_ga(T31, X133))
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)
GOAL1_IN_G(s(T11)) → U9_G(T11, s2tc9_in_ga(T11, T13))
U9_G(T11, s2tc9_out_ga(T11, T13)) → U10_G(T11, tappend10_in_gaaa(T13, X30, X4, X5))
U9_G(T11, s2tc9_out_ga(T11, T13)) → TAPPEND10_IN_GAAA(T13, X30, X4, X5)
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → U6_GAAA(T50, X274, X275, X276, tappend46_in_gaa(T50, X275, X276))
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → TAPPEND46_IN_GAA(T50, X275, X276)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → U4_GAA(T86, T84, T85, X371, X372, tappend46_in_gaa(T86, X371, X372))
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → U5_GAA(T93, T94, T96, X401, X402, tappend46_in_gaa(T96, X401, X402))
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → U7_GAAA(T101, X437, X438, X439, tappend46_in_gaa(T101, X438, X439))
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → TAPPEND46_IN_GAA(T101, X438, X439)
GOAL1_IN_G(s(T106)) → U11_G(T106, s2t9_in_ga(T106, X468))
GOAL1_IN_G(s(T106)) → U12_G(T106, s2tc9_in_ga(T106, T108))
U12_G(T106, s2tc9_out_ga(T106, T108)) → U13_G(T106, tappend46_in_gaa(node(nil, X467, T108), X4, X5))
U12_G(T106, s2tc9_out_ga(T106, T108)) → TAPPEND46_IN_GAA(node(nil, X467, T108), X4, X5)
GOAL1_IN_G(s(T115)) → U14_G(T115, s2t9_in_ga(T115, X512))
GOAL1_IN_G(s(T115)) → U15_G(T115, s2tc9_in_ga(T115, T117))
U15_G(T115, s2tc9_out_ga(T115, T117)) → U16_G(T115, tappend46_in_gaa(node(T117, X513, nil), X4, X5))
U15_G(T115, s2tc9_out_ga(T115, T117)) → TAPPEND46_IN_GAA(node(T117, X513, nil), X4, X5)
GOAL1_IN_G(s(T123)) → U17_G(T123, tappend10_in_gaaa(nil, X553, X4, X5))
GOAL1_IN_G(s(T123)) → TAPPEND10_IN_GAAA(nil, X553, X4, X5)
GOAL1_IN_G(0) → U18_G(tappend46_in_gaa(nil, X4, X5))
GOAL1_IN_G(0) → TAPPEND46_IN_GAA(nil, X4, X5)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2t9_in_ga(x1, x2)  =  s2t9_in_ga(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
tappend10_in_gaaa(x1, x2, x3, x4)  =  tappend10_in_gaaa(x1)
tappend46_in_gaa(x1, x2, x3)  =  tappend46_in_gaa(x1)
nil  =  nil
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x1, x2)
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)
TAPPEND10_IN_GAAA(x1, x2, x3, x4)  =  TAPPEND10_IN_GAAA(x1)
U6_GAAA(x1, x2, x3, x4, x5)  =  U6_GAAA(x1, x5)
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x1, x3, x6)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, x3, x6)
U7_GAAA(x1, x2, x3, x4, x5)  =  U7_GAAA(x1, x5)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1)  =  U18_G(x1)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 26 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
nil  =  nil
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)

R is empty.
The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x3)
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

TAPPEND46_IN_GAA(node(T93, T96)) → TAPPEND46_IN_GAA(T96)
TAPPEND46_IN_GAA(node(T86, T85)) → TAPPEND46_IN_GAA(T86)

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

(12) 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:

  • TAPPEND46_IN_GAA(node(T93, T96)) → TAPPEND46_IN_GAA(T96)
    The graph contains the following edges 1 > 1

  • TAPPEND46_IN_GAA(node(T86, T85)) → TAPPEND46_IN_GAA(T86)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
nil  =  nil
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

S2T9_IN_GA(s(T25)) → S2T9_IN_GA(T25)

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

(19) 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:

  • S2T9_IN_GA(s(T25)) → S2T9_IN_GA(T25)
    The graph contains the following edges 1 > 1

(20) YES