Left Termination proof of /tmp/tmpplVlnx/times.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

times₃(X, Y, Z) :- mult₄(X, Y, 0₀, Z).
mult₄(0₀, Y, 0₀, 0₀).
mult₄(s₁(U), Y, 0₀, Z) :- mult₄(U, Y, Y, Z).
mult₄(X, Y, s₁(W), s₁(Z)) :- mult₄(X, Y, W, Z).

With regard to the inferred argument filtering the predicates were used in the following modes:
times₃: (b,b,f)
mult₄: (b,b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

times_3_in_gga₃(X, Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
mult_4_in_ggga₄(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga₄(0_0, Y, 0_0, 0_0)
mult_4_in_ggga₄(s_1₁(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
mult_4_in_ggga₄(X, Y, s_1₁(W), s_1₁(Z)) -> if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_out_ggga₄(X, Y, W, Z)) -> mult_4_out_ggga₄(X, Y, s_1₁(W), s_1₁(Z))
if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_out_ggga₄(U, Y, Y, Z)) -> mult_4_out_ggga₄(s_1₁(U), Y, 0_0, Z)
if_times_3_in_1_gga₄(X, Y, Z, mult_4_out_ggga₄(X, Y, 0_0, Z)) -> times_3_out_gga₃(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:

times_3_in_gga₃(X, Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
mult_4_in_ggga₄(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga₄(0_0, Y, 0_0, 0_0)
mult_4_in_ggga₄(s_1₁(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
mult_4_in_ggga₄(X, Y, s_1₁(W), s_1₁(Z)) -> if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_out_ggga₄(X, Y, W, Z)) -> mult_4_out_ggga₄(X, Y, s_1₁(W), s_1₁(Z))
if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_out_ggga₄(U, Y, Y, Z)) -> mult_4_out_ggga₄(s_1₁(U), Y, 0_0, Z)
if_times_3_in_1_gga₄(X, Y, Z, mult_4_out_ggga₄(X, Y, 0_0, Z)) -> times_3_out_gga₃(X, Y, Z)

TIMES_3_IN_GGA₃(X, Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
TIMES_3_IN_GGA₃(X, Y, Z) -> MULT_4_IN_GGGA₄(X, Y, 0_0, Z)
MULT_4_IN_GGGA₄(s_1₁(U), Y, 0_0, Z) -> IF_MULT_4_IN_1_GGGA₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
MULT_4_IN_GGGA₄(s_1₁(U), Y, 0_0, Z) -> MULT_4_IN_GGGA₄(U, Y, Y, Z)
MULT_4_IN_GGGA₄(X, Y, s_1₁(W), s_1₁(Z)) -> IF_MULT_4_IN_2_GGGA₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
MULT_4_IN_GGGA₄(X, Y, s_1₁(W), s_1₁(Z)) -> MULT_4_IN_GGGA₄(X, Y, W, Z)

The TRS R consists of the following rules:

times_3_in_gga₃(X, Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
mult_4_in_ggga₄(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga₄(0_0, Y, 0_0, 0_0)
mult_4_in_ggga₄(s_1₁(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
mult_4_in_ggga₄(X, Y, s_1₁(W), s_1₁(Z)) -> if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_out_ggga₄(X, Y, W, Z)) -> mult_4_out_ggga₄(X, Y, s_1₁(W), s_1₁(Z))
if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_out_ggga₄(U, Y, Y, Z)) -> mult_4_out_ggga₄(s_1₁(U), Y, 0_0, Z)
if_times_3_in_1_gga₄(X, Y, Z, mult_4_out_ggga₄(X, Y, 0_0, Z)) -> times_3_out_gga₃(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga₃(x1, x2, x3) = times_3_in_gga₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
if_times_3_in_1_gga₄(x1, x2, x3, x4) = if_times_3_in_1_gga₁(x4)
mult_4_in_ggga₄(x1, x2, x3, x4) = mult_4_in_ggga₃(x1, x2, x3)
mult_4_out_ggga₄(x1, x2, x3, x4) = mult_4_out_ggga₁(x4)
if_mult_4_in_1_ggga₄(x1, x2, x3, x4) = if_mult_4_in_1_ggga₁(x4)
if_mult_4_in_2_ggga₅(x1, x2, x3, x4, x5) = if_mult_4_in_2_ggga₁(x5)
times_3_out_gga₃(x1, x2, x3) = times_3_out_gga₁(x3)
IF_TIMES_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_GGA₁(x4)
MULT_4_IN_GGGA₄(x1, x2, x3, x4) = MULT_4_IN_GGGA₃(x1, x2, x3)
TIMES_3_IN_GGA₃(x1, x2, x3) = TIMES_3_IN_GGA₂(x1, x2)
IF_MULT_4_IN_1_GGGA₄(x1, x2, x3, x4) = IF_MULT_4_IN_1_GGGA₁(x4)
IF_MULT_4_IN_2_GGGA₅(x1, x2, x3, x4, x5) = IF_MULT_4_IN_2_GGGA₁(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:

TIMES_3_IN_GGA₃(X, Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
TIMES_3_IN_GGA₃(X, Y, Z) -> MULT_4_IN_GGGA₄(X, Y, 0_0, Z)
MULT_4_IN_GGGA₄(s_1₁(U), Y, 0_0, Z) -> IF_MULT_4_IN_1_GGGA₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
MULT_4_IN_GGGA₄(s_1₁(U), Y, 0_0, Z) -> MULT_4_IN_GGGA₄(U, Y, Y, Z)
MULT_4_IN_GGGA₄(X, Y, s_1₁(W), s_1₁(Z)) -> IF_MULT_4_IN_2_GGGA₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
MULT_4_IN_GGGA₄(X, Y, s_1₁(W), s_1₁(Z)) -> MULT_4_IN_GGGA₄(X, Y, W, Z)

The TRS R consists of the following rules:

times_3_in_gga₃(X, Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
mult_4_in_ggga₄(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga₄(0_0, Y, 0_0, 0_0)
mult_4_in_ggga₄(s_1₁(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
mult_4_in_ggga₄(X, Y, s_1₁(W), s_1₁(Z)) -> if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_out_ggga₄(X, Y, W, Z)) -> mult_4_out_ggga₄(X, Y, s_1₁(W), s_1₁(Z))
if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_out_ggga₄(U, Y, Y, Z)) -> mult_4_out_ggga₄(s_1₁(U), Y, 0_0, Z)
if_times_3_in_1_gga₄(X, Y, Z, mult_4_out_ggga₄(X, Y, 0_0, Z)) -> times_3_out_gga₃(X, Y, Z)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

MULT_4_IN_GGGA₄(s_1₁(U), Y, 0_0, Z) -> MULT_4_IN_GGGA₄(U, Y, Y, Z)
MULT_4_IN_GGGA₄(X, Y, s_1₁(W), s_1₁(Z)) -> MULT_4_IN_GGGA₄(X, Y, W, Z)

The TRS R consists of the following rules:

times_3_in_gga₃(X, Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, mult_4_in_ggga₄(X, Y, 0_0, Z))
mult_4_in_ggga₄(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga₄(0_0, Y, 0_0, 0_0)
mult_4_in_ggga₄(s_1₁(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_in_ggga₄(U, Y, Y, Z))
mult_4_in_ggga₄(X, Y, s_1₁(W), s_1₁(Z)) -> if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_in_ggga₄(X, Y, W, Z))
if_mult_4_in_2_ggga₅(X, Y, W, Z, mult_4_out_ggga₄(X, Y, W, Z)) -> mult_4_out_ggga₄(X, Y, s_1₁(W), s_1₁(Z))
if_mult_4_in_1_ggga₄(U, Y, Z, mult_4_out_ggga₄(U, Y, Y, Z)) -> mult_4_out_ggga₄(s_1₁(U), Y, 0_0, Z)
if_times_3_in_1_gga₄(X, Y, Z, mult_4_out_ggga₄(X, Y, 0_0, Z)) -> times_3_out_gga₃(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga₃(x1, x2, x3) = times_3_in_gga₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
if_times_3_in_1_gga₄(x1, x2, x3, x4) = if_times_3_in_1_gga₁(x4)
mult_4_in_ggga₄(x1, x2, x3, x4) = mult_4_in_ggga₃(x1, x2, x3)
mult_4_out_ggga₄(x1, x2, x3, x4) = mult_4_out_ggga₁(x4)
if_mult_4_in_1_ggga₄(x1, x2, x3, x4) = if_mult_4_in_1_ggga₁(x4)
if_mult_4_in_2_ggga₅(x1, x2, x3, x4, x5) = if_mult_4_in_2_ggga₁(x5)
times_3_out_gga₃(x1, x2, x3) = times_3_out_gga₁(x3)
MULT_4_IN_GGGA₄(x1, x2, x3, x4) = MULT_4_IN_GGGA₃(x1, x2, x3)

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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

MULT_4_IN_GGGA₄(s_1₁(U), Y, 0_0, Z) -> MULT_4_IN_GGGA₄(U, Y, Y, Z)
MULT_4_IN_GGGA₄(X, Y, s_1₁(W), s_1₁(Z)) -> MULT_4_IN_GGGA₄(X, Y, W, Z)

R is empty.
The argument filtering Pi contains the following mapping:
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
MULT_4_IN_GGGA₄(x1, x2, x3, x4) = MULT_4_IN_GGGA₃(x1, x2, x3)

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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

MULT_4_IN_GGGA₃(s_1₁(U), Y, 0_0) -> MULT_4_IN_GGGA₃(U, Y, Y)
MULT_4_IN_GGGA₃(X, Y, s_1₁(W)) -> MULT_4_IN_GGGA₃(X, Y, W)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MULT_4_IN_GGGA₃}.
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_4_IN_GGGA₃(s_1₁(U), Y, 0_0) -> MULT_4_IN_GGGA₃(U, Y, Y)
The graph contains the following edges 1 > 1, 2 >= 2, 2 >= 3
MULT_4_IN_GGGA₃(X, Y, s_1₁(W)) -> MULT_4_IN_GGGA₃(X, Y, W)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3