Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

div(X, Y, Z) :- quot(X, Y, Y, Z).
quot(0, s(Y), s(Z), 0).
quot(s(X), s(Y), Z, U) :- quot(X, Y, Z, U).
quot(X, 0, s(Z), s(U)) :- quot(X, s(Z), s(Z), U).

Queries:

div(g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
div_in: (b,b,f)
quot_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:

div_in_gga(X, Y, Z) → U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
quot_in_ggga(0, s(Y), s(Z), 0) → quot_out_ggga(0, s(Y), s(Z), 0)
quot_in_ggga(s(X), s(Y), Z, U) → U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
quot_in_ggga(X, 0, s(Z), s(U)) → U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) → quot_out_ggga(X, 0, s(Z), s(U))
U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) → quot_out_ggga(s(X), s(Y), Z, U)
U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x4)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
div_out_gga(x1, x2, x3)  =  div_out_gga(x3)

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:

div_in_gga(X, Y, Z) → U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
quot_in_ggga(0, s(Y), s(Z), 0) → quot_out_ggga(0, s(Y), s(Z), 0)
quot_in_ggga(s(X), s(Y), Z, U) → U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
quot_in_ggga(X, 0, s(Z), s(U)) → U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) → quot_out_ggga(X, 0, s(Z), s(U))
U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) → quot_out_ggga(s(X), s(Y), Z, U)
U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x4)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
div_out_gga(x1, x2, x3)  =  div_out_gga(x3)


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

DIV_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
DIV_IN_GGA(X, Y, Z) → QUOT_IN_GGGA(X, Y, Y, Z)
QUOT_IN_GGGA(s(X), s(Y), Z, U) → U2_GGGA(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
QUOT_IN_GGGA(s(X), s(Y), Z, U) → QUOT_IN_GGGA(X, Y, Z, U)
QUOT_IN_GGGA(X, 0, s(Z), s(U)) → U3_GGGA(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
QUOT_IN_GGGA(X, 0, s(Z), s(U)) → QUOT_IN_GGGA(X, s(Z), s(Z), U)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
quot_in_ggga(0, s(Y), s(Z), 0) → quot_out_ggga(0, s(Y), s(Z), 0)
quot_in_ggga(s(X), s(Y), Z, U) → U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
quot_in_ggga(X, 0, s(Z), s(U)) → U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) → quot_out_ggga(X, 0, s(Z), s(U))
U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) → quot_out_ggga(s(X), s(Y), Z, U)
U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x4)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x4)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x5)

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:

DIV_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
DIV_IN_GGA(X, Y, Z) → QUOT_IN_GGGA(X, Y, Y, Z)
QUOT_IN_GGGA(s(X), s(Y), Z, U) → U2_GGGA(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
QUOT_IN_GGGA(s(X), s(Y), Z, U) → QUOT_IN_GGGA(X, Y, Z, U)
QUOT_IN_GGGA(X, 0, s(Z), s(U)) → U3_GGGA(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
QUOT_IN_GGGA(X, 0, s(Z), s(U)) → QUOT_IN_GGGA(X, s(Z), s(Z), U)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
quot_in_ggga(0, s(Y), s(Z), 0) → quot_out_ggga(0, s(Y), s(Z), 0)
quot_in_ggga(s(X), s(Y), Z, U) → U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
quot_in_ggga(X, 0, s(Z), s(U)) → U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) → quot_out_ggga(X, 0, s(Z), s(U))
U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) → quot_out_ggga(s(X), s(Y), Z, U)
U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x4)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x4)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 4 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

QUOT_IN_GGGA(X, 0, s(Z), s(U)) → QUOT_IN_GGGA(X, s(Z), s(Z), U)
QUOT_IN_GGGA(s(X), s(Y), Z, U) → QUOT_IN_GGGA(X, Y, Z, U)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
quot_in_ggga(0, s(Y), s(Z), 0) → quot_out_ggga(0, s(Y), s(Z), 0)
quot_in_ggga(s(X), s(Y), Z, U) → U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
quot_in_ggga(X, 0, s(Z), s(U)) → U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) → quot_out_ggga(X, 0, s(Z), s(U))
U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) → quot_out_ggga(s(X), s(Y), Z, U)
U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x4)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

QUOT_IN_GGGA(X, 0, s(Z), s(U)) → QUOT_IN_GGGA(X, s(Z), s(Z), U)
QUOT_IN_GGGA(s(X), s(Y), Z, U) → QUOT_IN_GGGA(X, Y, Z, U)

R is empty.
The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)

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
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

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

QUOT_IN_GGGA(X, 0, s(Z)) → QUOT_IN_GGGA(X, s(Z), s(Z))
QUOT_IN_GGGA(s(X), s(Y), Z) → QUOT_IN_GGGA(X, Y, Z)

R is empty.
Q is empty.
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: