Left Termination proof of /tmp/tmpYDrpTm/ackermann.pl

Left Termination of the query pattern ackermann_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

ackermann(0, N, s(N)).
ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val).
ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val)).

Queries:

ackermann(g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
ackermann_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ackermann_in_gga(0, N, s(N)) → ackermann_out_gga(0, N, s(N))
ackermann_in_gga(s(M), 0, Val) → U1_gga(M, Val, ackermann_in_gga(M, s(0), Val))
ackermann_in_gga(s(M), s(N), Val) → U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1))
U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val))
U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) → ackermann_out_gga(s(M), s(N), Val)
U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) → ackermann_out_gga(s(M), 0, Val)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

ackermann_in_gga(0, N, s(N)) → ackermann_out_gga(0, N, s(N))
ackermann_in_gga(s(M), 0, Val) → U1_gga(M, Val, ackermann_in_gga(M, s(0), Val))
ackermann_in_gga(s(M), s(N), Val) → U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1))
U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val))
U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) → ackermann_out_gga(s(M), s(N), Val)
U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) → ackermann_out_gga(s(M), 0, Val)

The argument filtering Pi contains the following mapping:
ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2)
0 = 0
ackermann_out_gga(x1, x2, x3) = ackermann_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)

Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

ACKERMANN_IN_GGA(s(M), 0, Val) → U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val))
ACKERMANN_IN_GGA(s(M), 0, Val) → ACKERMANN_IN_GGA(M, s(0), Val)
ACKERMANN_IN_GGA(s(M), s(N), Val) → U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1))
ACKERMANN_IN_GGA(s(M), s(N), Val) → ACKERMANN_IN_GGA(s(M), N, Val1)
U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val))
U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → ACKERMANN_IN_GGA(M, Val1, Val)

The TRS R consists of the following rules:

ackermann_in_gga(0, N, s(N)) → ackermann_out_gga(0, N, s(N))
ackermann_in_gga(s(M), 0, Val) → U1_gga(M, Val, ackermann_in_gga(M, s(0), Val))
ackermann_in_gga(s(M), s(N), Val) → U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1))
U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val))
U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) → ackermann_out_gga(s(M), s(N), Val)
U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) → ackermann_out_gga(s(M), 0, Val)

The argument filtering Pi contains the following mapping:
ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2)
0 = 0
ackermann_out_gga(x1, x2, x3) = ackermann_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)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4)
U1_GGA(x1, x2, x3) = U1_GGA(x3)
ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

ACKERMANN_IN_GGA(s(M), 0, Val) → U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val))
ACKERMANN_IN_GGA(s(M), 0, Val) → ACKERMANN_IN_GGA(M, s(0), Val)
ACKERMANN_IN_GGA(s(M), s(N), Val) → U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1))
ACKERMANN_IN_GGA(s(M), s(N), Val) → ACKERMANN_IN_GGA(s(M), N, Val1)
U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val))
U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → ACKERMANN_IN_GGA(M, Val1, Val)

The TRS R consists of the following rules:

ackermann_in_gga(0, N, s(N)) → ackermann_out_gga(0, N, s(N))
ackermann_in_gga(s(M), 0, Val) → U1_gga(M, Val, ackermann_in_gga(M, s(0), Val))
ackermann_in_gga(s(M), s(N), Val) → U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1))
U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val))
U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) → ackermann_out_gga(s(M), s(N), Val)
U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) → ackermann_out_gga(s(M), 0, Val)

The argument filtering Pi contains the following mapping:
ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2)
0 = 0
ackermann_out_gga(x1, x2, x3) = ackermann_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)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4)
U1_GGA(x1, x2, x3) = U1_GGA(x3)
ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ PiDPToQDPProof

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

U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → ACKERMANN_IN_GGA(M, Val1, Val)
ACKERMANN_IN_GGA(s(M), s(N), Val) → ACKERMANN_IN_GGA(s(M), N, Val1)
ACKERMANN_IN_GGA(s(M), 0, Val) → ACKERMANN_IN_GGA(M, s(0), Val)
ACKERMANN_IN_GGA(s(M), s(N), Val) → U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1))

The TRS R consists of the following rules:

ackermann_in_gga(0, N, s(N)) → ackermann_out_gga(0, N, s(N))
ackermann_in_gga(s(M), 0, Val) → U1_gga(M, Val, ackermann_in_gga(M, s(0), Val))
ackermann_in_gga(s(M), s(N), Val) → U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1))
U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) → U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val))
U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) → ackermann_out_gga(s(M), s(N), Val)
U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) → ackermann_out_gga(s(M), 0, Val)

The argument filtering Pi contains the following mapping:
ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2)
0 = 0
ackermann_out_gga(x1, x2, x3) = ackermann_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)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4)
ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ PiDPToQDPProof

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

ACKERMANN_IN_GGA(s(M), s(N)) → U2_GGA(M, ackermann_in_gga(s(M), N))
U2_GGA(M, ackermann_out_gga(Val1)) → ACKERMANN_IN_GGA(M, Val1)
ACKERMANN_IN_GGA(s(M), 0) → ACKERMANN_IN_GGA(M, s(0))
ACKERMANN_IN_GGA(s(M), s(N)) → ACKERMANN_IN_GGA(s(M), N)

The TRS R consists of the following rules:

ackermann_in_gga(0, N) → ackermann_out_gga(s(N))
ackermann_in_gga(s(M), 0) → U1_gga(ackermann_in_gga(M, s(0)))
ackermann_in_gga(s(M), s(N)) → U2_gga(M, ackermann_in_gga(s(M), N))
U2_gga(M, ackermann_out_gga(Val1)) → U3_gga(ackermann_in_gga(M, Val1))
U3_gga(ackermann_out_gga(Val)) → ackermann_out_gga(Val)
U1_gga(ackermann_out_gga(Val)) → ackermann_out_gga(Val)

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

ACKERMANN_IN_GGA(s(M), s(N)) → U2_GGA(M, ackermann_in_gga(s(M), N))
The graph contains the following edges 1 > 1
U2_GGA(M, ackermann_out_gga(Val1)) → ACKERMANN_IN_GGA(M, Val1)
The graph contains the following edges 1 >= 1, 2 > 2
ACKERMANN_IN_GGA(s(M), s(N)) → ACKERMANN_IN_GGA(s(M), N)
The graph contains the following edges 1 >= 1, 2 > 2
ACKERMANN_IN_GGA(s(M), 0) → ACKERMANN_IN_GGA(M, s(0))
The graph contains the following edges 1 > 1