(0) Obligation:

Clauses:

isNat(s(X)) :- isNat(X).
isNat(0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).
lt(s(X), s(Y)) :- lt(X, Y).
lt(0, s(Y)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
even(s(X)) :- odd(X).
even(0).
odd(s(X)) :- even(X).
odd(s(0)).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).
mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)).
mult(0, Y, 0).
factorial(s(X), R) :- ','(factorial(X, Y), mult(s(X), Y, R)).
factorial(0, s(0)).

Queries:

factorial(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
FACTORIAL_IN_GA(x1, x2)  =  FACTORIAL_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)

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

(4) Obligation:

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

FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
FACTORIAL_IN_GA(x1, x2)  =  FACTORIAL_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

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

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)
    The graph contains the following edges 1 > 1, 2 >= 2

(13) TRUE

(14) Obligation:

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

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

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

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

MULT_IN_GGA(s(X), Y) → MULT_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GGA(s(X), Y) → MULT_IN_GGA(X, Y)
    The graph contains the following edges 1 > 1, 2 >= 2

(20) TRUE

(21) Obligation:

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

FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
FACTORIAL_IN_GA(x1, x2)  =  FACTORIAL_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)

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

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

FACTORIAL_IN_GA(s(X)) → FACTORIAL_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GA(s(X)) → FACTORIAL_IN_GA(X)
    The graph contains the following edges 1 > 1

(27) TRUE

(28) PrologToPiTRSProof (SOUND transformation)

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

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(29) Obligation:

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

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

(30) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
FACTORIAL_IN_GA(x1, x2)  =  FACTORIAL_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x3)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x4)

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

(31) Obligation:

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

FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
FACTORIAL_IN_GA(x1, x2)  =  FACTORIAL_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x3)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x4)

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

(32) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(33) Complex Obligation (AND)

(34) Obligation:

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

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(35) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(36) Obligation:

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

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

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

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

(37) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(38) Obligation:

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

ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(39) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)
    The graph contains the following edges 1 > 1, 2 >= 2

(40) TRUE

(41) Obligation:

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

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

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

(42) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(43) Obligation:

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

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

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

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

(44) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(45) Obligation:

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

MULT_IN_GGA(s(X), Y) → MULT_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(46) Obligation:

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

FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)

The TRS R consists of the following rules:

factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)

The argument filtering Pi contains the following mapping:
factorial_in_ga(x1, x2)  =  factorial_in_ga(x1)
s(x1)  =  s(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
0  =  0
factorial_out_ga(x1, x2)  =  factorial_out_ga(x2)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
FACTORIAL_IN_GA(x1, x2)  =  FACTORIAL_IN_GA(x1)

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

(47) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(48) Obligation:

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

FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)

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

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