Left Termination of the query pattern ack_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 PiDPToQDPProof (⇐)
    ↳8 QDP
    ↳9 QDPSizeChangeProof (⇔)
    ↳10 TRUE
  ↳11 PrologToPiTRSProof (⇐)
    ↳12 PiTRS
    ↳13 DependencyPairsProof (⇔)
    ↳14 PiDP
    ↳15 DependencyGraphProof (⇔)
    ↳16 PiDP
    ↳17 PiDPToQDPProof (⇐)
    ↳18 QDP

(0) Obligation:

Clauses:

ack(0, N, s(N)).
ack(s(M), 0, A) :- ack(M, s(0), A).
ack(s(M), s(N), A) :- ','(ack(s(M), N, A1), ack(M, A1, A)).

Queries:

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

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(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:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(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:

ACK_IN_GGA(s(M), 0, A) → U1_GGA(M, A, ack_in_gga(M, s(0), A))
ACK_IN_GGA(s(M), 0, A) → ACK_IN_GGA(M, s(0), A)
ACK_IN_GGA(s(M), s(N), A) → U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
ACK_IN_GGA(s(M), s(N), A) → ACK_IN_GGA(s(M), N, A1)
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → U3_GGA(M, N, A, ack_in_gga(M, A1, A))
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1, A)

The TRS R consists of the following rules:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

(4) Obligation:

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

ACK_IN_GGA(s(M), 0, A) → U1_GGA(M, A, ack_in_gga(M, s(0), A))
ACK_IN_GGA(s(M), 0, A) → ACK_IN_GGA(M, s(0), A)
ACK_IN_GGA(s(M), s(N), A) → U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
ACK_IN_GGA(s(M), s(N), A) → ACK_IN_GGA(s(M), N, A1)
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → U3_GGA(M, N, A, ack_in_gga(M, A1, A))
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1, A)

The TRS R consists of the following rules:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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:

ACK_IN_GGA(s(M), 0, A) → ACK_IN_GGA(M, s(0), A)
ACK_IN_GGA(s(M), s(N), A) → U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1, A)
ACK_IN_GGA(s(M), s(N), A) → ACK_IN_GGA(s(M), N, A1)

The TRS R consists of the following rules:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

ACK_IN_GGA(s(M), 0) → ACK_IN_GGA(M, s(0))
ACK_IN_GGA(s(M), s(N)) → U2_GGA(M, ack_in_gga(s(M), N))
U2_GGA(M, ack_out_gga(A1)) → ACK_IN_GGA(M, A1)
ACK_IN_GGA(s(M), s(N)) → ACK_IN_GGA(s(M), N)

The TRS R consists of the following rules:

ack_in_gga(0, N) → ack_out_gga(s(N))
ack_in_gga(s(M), 0) → U1_gga(ack_in_gga(M, s(0)))
ack_in_gga(s(M), s(N)) → U2_gga(M, ack_in_gga(s(M), N))
U2_gga(M, ack_out_gga(A1)) → U3_gga(ack_in_gga(M, A1))
U3_gga(ack_out_gga(A)) → ack_out_gga(A)
U1_gga(ack_out_gga(A)) → ack_out_gga(A)

The set Q consists of the following terms:

ack_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0)
U1_gga(x0)

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

(9) 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:

  • ACK_IN_GGA(s(M), s(N)) → ACK_IN_GGA(s(M), N)
    The graph contains the following edges 1 >= 1, 2 > 2

  • ACK_IN_GGA(s(M), s(N)) → U2_GGA(M, ack_in_gga(s(M), N))
    The graph contains the following edges 1 > 1

  • U2_GGA(M, ack_out_gga(A1)) → ACK_IN_GGA(M, A1)
    The graph contains the following edges 1 >= 1, 2 > 2

  • ACK_IN_GGA(s(M), 0) → ACK_IN_GGA(M, s(0))
    The graph contains the following edges 1 > 1

(10) TRUE

(11) PrologToPiTRSProof (SOUND transformation)

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

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) Obligation:

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

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)

(13) 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:

ACK_IN_GGA(s(M), 0, A) → U1_GGA(M, A, ack_in_gga(M, s(0), A))
ACK_IN_GGA(s(M), 0, A) → ACK_IN_GGA(M, s(0), A)
ACK_IN_GGA(s(M), s(N), A) → U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
ACK_IN_GGA(s(M), s(N), A) → ACK_IN_GGA(s(M), N, A1)
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → U3_GGA(M, N, A, ack_in_gga(M, A1, A))
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1, A)

The TRS R consists of the following rules:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(14) Obligation:

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

ACK_IN_GGA(s(M), 0, A) → U1_GGA(M, A, ack_in_gga(M, s(0), A))
ACK_IN_GGA(s(M), 0, A) → ACK_IN_GGA(M, s(0), A)
ACK_IN_GGA(s(M), s(N), A) → U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
ACK_IN_GGA(s(M), s(N), A) → ACK_IN_GGA(s(M), N, A1)
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → U3_GGA(M, N, A, ack_in_gga(M, A1, A))
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1, A)

The TRS R consists of the following rules:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

ACK_IN_GGA(s(M), 0, A) → ACK_IN_GGA(M, s(0), A)
ACK_IN_GGA(s(M), s(N), A) → U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1, A)
ACK_IN_GGA(s(M), s(N), A) → ACK_IN_GGA(s(M), N, A1)

The TRS R consists of the following rules:

ack_in_gga(0, N, s(N)) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0, A) → U1_gga(M, A, ack_in_gga(M, s(0), A))
ack_in_gga(s(M), s(N), A) → U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, A, ack_in_gga(M, A1, A))
U3_gga(M, N, A, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, A, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The argument filtering Pi contains the following mapping:
ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
0  =  0
ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)

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:

ACK_IN_GGA(s(M), 0) → ACK_IN_GGA(M, s(0))
ACK_IN_GGA(s(M), s(N)) → U2_GGA(M, N, ack_in_gga(s(M), N))
U2_GGA(M, N, ack_out_gga(s(M), N, A1)) → ACK_IN_GGA(M, A1)
ACK_IN_GGA(s(M), s(N)) → ACK_IN_GGA(s(M), N)

The TRS R consists of the following rules:

ack_in_gga(0, N) → ack_out_gga(0, N, s(N))
ack_in_gga(s(M), 0) → U1_gga(M, ack_in_gga(M, s(0)))
ack_in_gga(s(M), s(N)) → U2_gga(M, N, ack_in_gga(s(M), N))
U2_gga(M, N, ack_out_gga(s(M), N, A1)) → U3_gga(M, N, ack_in_gga(M, A1))
U3_gga(M, N, ack_out_gga(M, A1, A)) → ack_out_gga(s(M), s(N), A)
U1_gga(M, ack_out_gga(M, s(0), A)) → ack_out_gga(s(M), 0, A)

The set Q consists of the following terms:

ack_in_gga(x0, x1)
U2_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U1_gga(x0, x1)

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