Left Termination of the query pattern f_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 QDPSizeChangeProof (⇔)
    ↳12 TRUE
  ↳13 PrologToPiTRSProof (⇐)
    ↳14 PiTRS
    ↳15 DependencyPairsProof (⇔)
    ↳16 PiDP
    ↳17 DependencyGraphProof (⇔)
    ↳18 PiDP
    ↳19 UsableRulesProof (⇔)
    ↳20 PiDP
    ↳21 PiDPToQDPProof (⇐)
    ↳22 QDP

(0) Obligation:

Clauses:

f([], RES, RES).
f(.(Head, Tail), X, RES) :- g(Tail, X, .(Head, Tail), RES).
g(A, B, C, RES) :- f(A, .(B, C), RES).

Queries:

f(g,g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)

(3) 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:

F_IN_GGA(.(Head, Tail), X, RES) → U1_GGA(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → U2_GGGA(A, B, C, RES, f_in_gga(A, .(B, C), RES))
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

The TRS R consists of the following rules:

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)

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

(4) Obligation:

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

F_IN_GGA(.(Head, Tail), X, RES) → U1_GGA(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → U2_GGGA(A, B, C, RES, f_in_gga(A, .(B, C), RES))
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

The TRS R consists of the following rules:

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(6) Obligation:

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

F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

The TRS R consists of the following rules:

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

F_IN_GGA(.(Head, Tail), X) → G_IN_GGGA(Tail, X, .(Head, Tail))
G_IN_GGGA(A, B, C) → F_IN_GGA(A, .(B, C))

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

(11) 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:

  • G_IN_GGGA(A, B, C) → F_IN_GGA(A, .(B, C))
    The graph contains the following edges 1 >= 1

  • F_IN_GGA(.(Head, Tail), X) → G_IN_GGGA(Tail, X, .(Head, Tail))
    The graph contains the following edges 1 > 1, 2 >= 2, 1 >= 3

(12) TRUE

(13) PrologToPiTRSProof (SOUND transformation)

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

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x4)

(15) 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:

F_IN_GGA(.(Head, Tail), X, RES) → U1_GGA(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → U2_GGGA(A, B, C, RES, f_in_gga(A, .(B, C), RES))
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

The TRS R consists of the following rules:

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x4)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x5)
G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x5)

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

(16) Obligation:

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

F_IN_GGA(.(Head, Tail), X, RES) → U1_GGA(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → U2_GGGA(A, B, C, RES, f_in_gga(A, .(B, C), RES))
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

The TRS R consists of the following rules:

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x4)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x5)
G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x5)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(18) Obligation:

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

F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

The TRS R consists of the following rules:

f_in_gga([], RES, RES) → f_out_gga([], RES, RES)
f_in_gga(.(Head, Tail), X, RES) → U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
g_in_ggga(A, B, C, RES) → U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) → g_out_ggga(A, B, C, RES)
U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) → f_out_gga(.(Head, Tail), X, RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x4)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

F_IN_GGA(.(Head, Tail), X, RES) → G_IN_GGGA(Tail, X, .(Head, Tail), RES)
G_IN_GGGA(A, B, C, RES) → F_IN_GGA(A, .(B, C), RES)

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

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

F_IN_GGA(.(Head, Tail), X) → G_IN_GGGA(Tail, X, .(Head, Tail))
G_IN_GGGA(A, B, C) → F_IN_GGA(A, .(B, C))

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