Left Termination proof of /tmp/tmp0XfHjn/pl7.2.9.pl

Left Termination of the query pattern mult(b,b,f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

mult₃(0₀, Y, 0₀).
mult₃(s₁(X), Y, Z) :- mult₃(X, Y, Z1), add₃(Z1, Y, Z).
add₃(0₀, Y, Y).
add₃(s₁(X), Y, s₁(Z)) :- add₃(X, Y, Z).

With regard to the inferred argument filtering the predicates were used in the following modes:
mult₃: (b,b,f)
add₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
mult_3_in_gga₃(s_1₁(X), Y, Z) -> if_mult_3_in_1_gga₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
if_mult_3_in_1_gga₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
add_3_in_gga₃(0_0, Y, Y) -> add_3_out_gga₃(0_0, Y, Y)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_out_gga₃(Z1, Y, Z)) -> mult_3_out_gga₃(s_1₁(X), Y, Z)

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:

mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
mult_3_in_gga₃(s_1₁(X), Y, Z) -> if_mult_3_in_1_gga₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
if_mult_3_in_1_gga₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
add_3_in_gga₃(0_0, Y, Y) -> add_3_out_gga₃(0_0, Y, Y)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_out_gga₃(Z1, Y, Z)) -> mult_3_out_gga₃(s_1₁(X), Y, Z)

MULT_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_MULT_3_IN_1_GGA₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
MULT_3_IN_GGA₃(s_1₁(X), Y, Z) -> MULT_3_IN_GGA₃(X, Y, Z1)
IF_MULT_3_IN_1_GGA₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> IF_MULT_3_IN_2_GGA₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
IF_MULT_3_IN_1_GGA₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> ADD_3_IN_GGA₃(Z1, Y, Z)
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GGA₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
mult_3_in_gga₃(s_1₁(X), Y, Z) -> if_mult_3_in_1_gga₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
if_mult_3_in_1_gga₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
add_3_in_gga₃(0_0, Y, Y) -> add_3_out_gga₃(0_0, Y, Y)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_out_gga₃(Z1, Y, Z)) -> mult_3_out_gga₃(s_1₁(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga₃(x1, x2, x3) = mult_3_in_gga₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
mult_3_out_gga₃(x1, x2, x3) = mult_3_out_gga₁(x3)
if_mult_3_in_1_gga₄(x1, x2, x3, x4) = if_mult_3_in_1_gga₂(x2, x4)
if_mult_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_mult_3_in_2_gga₁(x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
IF_ADD_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_ADD_3_IN_1_GGA₁(x4)
IF_MULT_3_IN_2_GGA₅(x1, x2, x3, x4, x5) = IF_MULT_3_IN_2_GGA₁(x5)
MULT_3_IN_GGA₃(x1, x2, x3) = MULT_3_IN_GGA₂(x1, x2)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_3_IN_GGA₂(x1, x2)
IF_MULT_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_MULT_3_IN_1_GGA₂(x2, 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:

MULT_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_MULT_3_IN_1_GGA₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
MULT_3_IN_GGA₃(s_1₁(X), Y, Z) -> MULT_3_IN_GGA₃(X, Y, Z1)
IF_MULT_3_IN_1_GGA₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> IF_MULT_3_IN_2_GGA₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
IF_MULT_3_IN_1_GGA₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> ADD_3_IN_GGA₃(Z1, Y, Z)
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GGA₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
mult_3_in_gga₃(s_1₁(X), Y, Z) -> if_mult_3_in_1_gga₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
if_mult_3_in_1_gga₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
add_3_in_gga₃(0_0, Y, Y) -> add_3_out_gga₃(0_0, Y, Y)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_out_gga₃(Z1, Y, Z)) -> mult_3_out_gga₃(s_1₁(X), Y, Z)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
mult_3_in_gga₃(s_1₁(X), Y, Z) -> if_mult_3_in_1_gga₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
if_mult_3_in_1_gga₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
add_3_in_gga₃(0_0, Y, Y) -> add_3_out_gga₃(0_0, Y, Y)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_out_gga₃(Z1, Y, Z)) -> mult_3_out_gga₃(s_1₁(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga₃(x1, x2, x3) = mult_3_in_gga₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
mult_3_out_gga₃(x1, x2, x3) = mult_3_out_gga₁(x3)
if_mult_3_in_1_gga₄(x1, x2, x3, x4) = if_mult_3_in_1_gga₂(x2, x4)
if_mult_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_mult_3_in_2_gga₁(x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_3_IN_GGA₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_3_IN_GGA₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

ADD_3_IN_GGA₂(s_1₁(X), Y) -> ADD_3_IN_GGA₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ADD_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis 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:

ADD_3_IN_GGA₂(s_1₁(X), Y) -> ADD_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

MULT_3_IN_GGA₃(s_1₁(X), Y, Z) -> MULT_3_IN_GGA₃(X, Y, Z1)

The TRS R consists of the following rules:

mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
mult_3_in_gga₃(s_1₁(X), Y, Z) -> if_mult_3_in_1_gga₄(X, Y, Z, mult_3_in_gga₃(X, Y, Z1))
if_mult_3_in_1_gga₄(X, Y, Z, mult_3_out_gga₃(X, Y, Z1)) -> if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_in_gga₃(Z1, Y, Z))
add_3_in_gga₃(0_0, Y, Y) -> add_3_out_gga₃(0_0, Y, Y)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_mult_3_in_2_gga₅(X, Y, Z, Z1, add_3_out_gga₃(Z1, Y, Z)) -> mult_3_out_gga₃(s_1₁(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga₃(x1, x2, x3) = mult_3_in_gga₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
mult_3_out_gga₃(x1, x2, x3) = mult_3_out_gga₁(x3)
if_mult_3_in_1_gga₄(x1, x2, x3, x4) = if_mult_3_in_1_gga₂(x2, x4)
if_mult_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_mult_3_in_2_gga₁(x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
MULT_3_IN_GGA₃(x1, x2, x3) = MULT_3_IN_GGA₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

MULT_3_IN_GGA₃(s_1₁(X), Y, Z) -> MULT_3_IN_GGA₃(X, Y, Z1)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
MULT_3_IN_GGA₃(x1, x2, x3) = MULT_3_IN_GGA₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

MULT_3_IN_GGA₂(s_1₁(X), Y) -> MULT_3_IN_GGA₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MULT_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis 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:

MULT_3_IN_GGA₂(s_1₁(X), Y) -> MULT_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2