Left Termination proof of /tmp/tmphRpw21/nat.pl

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

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

isNat₁(s₁(X)) :- isNat₁(X).
isNat₁(0₀).
notEq₂(s₁(X), s₁(Y)) :- notEq₂(X, Y).
notEq₂(s₁(X), 0₀).
notEq₂(0₀, s₁(X)).
lt₂(s₁(X), s₁(Y)) :- lt₂(X, Y).
lt₂(0₀, s₁(Y)).
gt₂(s₁(X), s₁(Y)) :- gt₂(X, Y).
gt₂(s₁(X), 0₀).
le₂(s₁(X), s₁(Y)) :- le₂(X, Y).
le₂(0₀, s₁(Y)).
le₂(0₀, 0₀).
even₁(s₁(X)) :- odd₁(X).
even₁(0₀).
odd₁(s₁(X)) :- even₁(X).
odd₁(s₁(0₀)).
add₃(s₁(X), Y, s₁(Z)) :- add₃(X, Y, Z).
add₃(0₀, X, X).
mult₃(s₁(X), Y, R) :- mult₃(X, Y, Z), add₃(Y, Z, R).
mult₃(0₀, Y, 0₀).
factorial₂(s₁(X), R) :- factorial₂(X, Y), mult₃(s₁(X), Y, R).
factorial₂(0₀, s₁(0₀)).

The clauses

isNat₁(s₁(X)) :- isNat₁(X).
isNat₁(0₀).
notEq₂(s₁(X), s₁(Y)) :- notEq₂(X, Y).
notEq₂(s₁(X), 0₀).
notEq₂(0₀, s₁(X)).
lt₂(s₁(X), s₁(Y)) :- lt₂(X, Y).
lt₂(0₀, s₁(Y)).
gt₂(s₁(X), s₁(Y)) :- gt₂(X, Y).
gt₂(s₁(X), 0₀).
le₂(s₁(X), s₁(Y)) :- le₂(X, Y).
le₂(0₀, s₁(Y)).
le₂(0₀, 0₀).
even₁(s₁(X)) :- odd₁(X).
even₁(0₀).
odd₁(s₁(X)) :- even₁(X).
odd₁(s₁(0₀)).

can be ignored, as they are not needed by any of the given querys.

Deleting these clauses results in the following prolog program:

add₃(s₁(X), Y, s₁(Z)) :- add₃(X, Y, Z).
add₃(0₀, X, X).
mult₃(s₁(X), Y, R) :- mult₃(X, Y, Z), add₃(Y, Z, R).
mult₃(0₀, Y, 0₀).
factorial₂(s₁(X), R) :- factorial₂(X, Y), mult₃(s₁(X), Y, R).
factorial₂(0₀, s₁(0₀)).

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

With regard to the inferred argument filtering the predicates were used in the following modes:
factorial₂: (b,f)
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:

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

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

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

FACTORIAL_2_IN_GA₂(s_1₁(X), R) -> IF_FACTORIAL_2_IN_1_GA₃(X, R, factorial_2_in_ga₂(X, Y))
FACTORIAL_2_IN_GA₂(s_1₁(X), R) -> FACTORIAL_2_IN_GA₂(X, Y)
IF_FACTORIAL_2_IN_1_GA₃(X, R, factorial_2_out_ga₂(X, Y)) -> IF_FACTORIAL_2_IN_2_GA₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
IF_FACTORIAL_2_IN_1_GA₃(X, R, factorial_2_out_ga₂(X, Y)) -> MULT_3_IN_GGA₃(s_1₁(X), Y, R)
MULT_3_IN_GGA₃(s_1₁(X), Y, R) -> IF_MULT_3_IN_1_GGA₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
MULT_3_IN_GGA₃(s_1₁(X), Y, R) -> MULT_3_IN_GGA₃(X, Y, Z)
IF_MULT_3_IN_1_GGA₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> IF_MULT_3_IN_2_GGA₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
IF_MULT_3_IN_1_GGA₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> ADD_3_IN_GGA₃(Y, Z, R)
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:

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

The argument filtering Pi contains the following mapping:
factorial_2_in_ga₂(x1, x2) = factorial_2_in_ga₁(x1)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
if_factorial_2_in_1_ga₃(x1, x2, x3) = if_factorial_2_in_1_ga₂(x1, x3)
factorial_2_out_ga₂(x1, x2) = factorial_2_out_ga₁(x2)
if_factorial_2_in_2_ga₄(x1, x2, x3, x4) = if_factorial_2_in_2_ga₁(x4)
mult_3_in_gga₃(x1, x2, x3) = mult_3_in_gga₂(x1, x2)
if_mult_3_in_1_gga₄(x1, x2, x3, x4) = if_mult_3_in_1_gga₂(x2, x4)
mult_3_out_gga₃(x1, x2, x3) = mult_3_out_gga₁(x3)
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)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
IF_FACTORIAL_2_IN_1_GA₃(x1, x2, x3) = IF_FACTORIAL_2_IN_1_GA₂(x1, x3)
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)
IF_FACTORIAL_2_IN_2_GA₄(x1, x2, x3, x4) = IF_FACTORIAL_2_IN_2_GA₁(x4)
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)
FACTORIAL_2_IN_GA₂(x1, x2) = FACTORIAL_2_IN_GA₁(x1)

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

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

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

FACTORIAL_2_IN_GA₂(s_1₁(X), R) -> IF_FACTORIAL_2_IN_1_GA₃(X, R, factorial_2_in_ga₂(X, Y))
FACTORIAL_2_IN_GA₂(s_1₁(X), R) -> FACTORIAL_2_IN_GA₂(X, Y)
IF_FACTORIAL_2_IN_1_GA₃(X, R, factorial_2_out_ga₂(X, Y)) -> IF_FACTORIAL_2_IN_2_GA₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
IF_FACTORIAL_2_IN_1_GA₃(X, R, factorial_2_out_ga₂(X, Y)) -> MULT_3_IN_GGA₃(s_1₁(X), Y, R)
MULT_3_IN_GGA₃(s_1₁(X), Y, R) -> IF_MULT_3_IN_1_GGA₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
MULT_3_IN_GGA₃(s_1₁(X), Y, R) -> MULT_3_IN_GGA₃(X, Y, Z)
IF_MULT_3_IN_1_GGA₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> IF_MULT_3_IN_2_GGA₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
IF_MULT_3_IN_1_GGA₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> ADD_3_IN_GGA₃(Y, Z, R)
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:

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ 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:

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ 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

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

                  ↳ 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

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

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

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

The TRS R consists of the following rules:

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

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

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

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

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

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

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

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

FACTORIAL_2_IN_GA₂(s_1₁(X), R) -> FACTORIAL_2_IN_GA₂(X, Y)

The TRS R consists of the following rules:

factorial_2_in_ga₂(s_1₁(X), R) -> if_factorial_2_in_1_ga₃(X, R, factorial_2_in_ga₂(X, Y))
factorial_2_in_ga₂(0_0, s_1₁(0_0)) -> factorial_2_out_ga₂(0_0, s_1₁(0_0))
if_factorial_2_in_1_ga₃(X, R, factorial_2_out_ga₂(X, Y)) -> if_factorial_2_in_2_ga₄(X, R, Y, mult_3_in_gga₃(s_1₁(X), Y, R))
mult_3_in_gga₃(s_1₁(X), Y, R) -> if_mult_3_in_1_gga₄(X, Y, R, mult_3_in_gga₃(X, Y, Z))
mult_3_in_gga₃(0_0, Y, 0_0) -> mult_3_out_gga₃(0_0, Y, 0_0)
if_mult_3_in_1_gga₄(X, Y, R, mult_3_out_gga₃(X, Y, Z)) -> if_mult_3_in_2_gga₅(X, Y, R, Z, add_3_in_gga₃(Y, Z, R))
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(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
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, R, Z, add_3_out_gga₃(Y, Z, R)) -> mult_3_out_gga₃(s_1₁(X), Y, R)
if_factorial_2_in_2_ga₄(X, R, Y, mult_3_out_gga₃(s_1₁(X), Y, R)) -> factorial_2_out_ga₂(s_1₁(X), R)

↳ PROLOG

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

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

FACTORIAL_2_IN_GA₂(s_1₁(X), R) -> FACTORIAL_2_IN_GA₂(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
FACTORIAL_2_IN_GA₂(x1, x2) = FACTORIAL_2_IN_GA₁(x1)

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

  ↳ UnrequestedClauseRemoverProof

    ↳ PROLOG

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

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

FACTORIAL_2_IN_GA₁(s_1₁(X)) -> FACTORIAL_2_IN_GA₁(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {FACTORIAL_2_IN_GA₁}.
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:

FACTORIAL_2_IN_GA₁(s_1₁(X)) -> FACTORIAL_2_IN_GA₁(X)
The graph contains the following edges 1 > 1