(0) Obligation:

Clauses:

mult(0, Y, 0).
mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z)).
add(0, Y, Y).
add(s(X), Y, s(Z)) :- add(X, Y, Z).

Queries:

mult(g,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:
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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

(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:

MULT_IN_GGA(s(X), Y, Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, Z1))
MULT_IN_GGA(s(X), Y, Z) → MULT_IN_GGA(X, Y, Z1)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_GGA(X, Y, Z, add_in_gga(Z1, Y, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → ADD_IN_GGA(Z1, Y, Z)
ADD_IN_GGA(s(X), Y, s(Z)) → U3_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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

(4) Obligation:

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

MULT_IN_GGA(s(X), Y, Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, Z1))
MULT_IN_GGA(s(X), Y, Z) → MULT_IN_GGA(X, Y, Z1)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_GGA(X, Y, Z, add_in_gga(Z1, Y, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → ADD_IN_GGA(Z1, Y, Z)
ADD_IN_GGA(s(X), Y, s(Z)) → U3_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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
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, Z) → MULT_IN_GGA(X, Y, Z1)

The TRS R consists of the following rules:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
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, Z) → MULT_IN_GGA(X, Y, Z1)

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) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)

(23) 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:

MULT_IN_GGA(s(X), Y, Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, Z1))
MULT_IN_GGA(s(X), Y, Z) → MULT_IN_GGA(X, Y, Z1)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_GGA(X, Y, Z, add_in_gga(Z1, Y, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → ADD_IN_GGA(Z1, Y, Z)
ADD_IN_GGA(s(X), Y, s(Z)) → U3_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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(24) Obligation:

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

MULT_IN_GGA(s(X), Y, Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, Z1))
MULT_IN_GGA(s(X), Y, Z) → MULT_IN_GGA(X, Y, Z1)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_GGA(X, Y, Z, add_in_gga(Z1, Y, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) → ADD_IN_GGA(Z1, Y, Z)
ADD_IN_GGA(s(X), Y, s(Z)) → U3_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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(26) Complex Obligation (AND)

(27) 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:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) 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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) 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.

(32) 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

(33) TRUE

(34) Obligation:

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

MULT_IN_GGA(s(X), Y, Z) → MULT_IN_GGA(X, Y, Z1)

The TRS R consists of the following rules:

mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
mult_in_gga(s(X), Y, Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) → U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
add_in_gga(0, Y, Y) → add_out_gga(0, Y, Y)
add_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) → mult_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
0  =  0
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_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:

MULT_IN_GGA(s(X), Y, Z) → MULT_IN_GGA(X, Y, Z1)

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

(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:

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.