(0) Obligation:
Clauses:
mult(X1, 0, 0).
mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)).
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(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)
sum_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(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_out_gga(
x1,
x2,
x3)
U3_gga(
x1,
x2,
x3,
x4) =
U3_gga(
x1,
x2,
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(X, s(Y), Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, W))
MULT_IN_GGA(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → U2_GGA(X, Y, Z, sum_in_gga(W, X, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → SUM_IN_GGA(W, X, Z)
SUM_IN_GGA(X, s(Y), s(Z)) → U3_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_IN_GGA(
x1,
x2)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_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:
MULT_IN_GGA(X, s(Y), Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, W))
MULT_IN_GGA(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → U2_GGA(X, Y, Z, sum_in_gga(W, X, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → SUM_IN_GGA(W, X, Z)
SUM_IN_GGA(X, s(Y), s(Z)) → U3_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_IN_GGA(
x1,
x2)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_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 2 SCCs with 4 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_out_gga(
x1,
x2,
x3)
U3_gga(
x1,
x2,
x3,
x4) =
U3_gga(
x1,
x2,
x4)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_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:
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_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:
SUM_IN_GGA(X, s(Y)) → SUM_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:
- SUM_IN_GGA(X, s(Y)) → SUM_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(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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
(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(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
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(X, s(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(X, s(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)
sum_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(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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(
x1,
x4)
U2_gga(
x1,
x2,
x3,
x4) =
U2_gga(
x4)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_out_gga(
x3)
U3_gga(
x1,
x2,
x3,
x4) =
U3_gga(
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(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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(
x1,
x4)
U2_gga(
x1,
x2,
x3,
x4) =
U2_gga(
x4)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_out_gga(
x3)
U3_gga(
x1,
x2,
x3,
x4) =
U3_gga(
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(X, s(Y), Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, W))
MULT_IN_GGA(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → U2_GGA(X, Y, Z, sum_in_gga(W, X, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → SUM_IN_GGA(W, X, Z)
SUM_IN_GGA(X, s(Y), s(Z)) → U3_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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(
x1,
x4)
U2_gga(
x1,
x2,
x3,
x4) =
U2_gga(
x4)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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(
x1,
x4)
U2_GGA(
x1,
x2,
x3,
x4) =
U2_GGA(
x4)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_IN_GGA(
x1,
x2)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_GGA(
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(X, s(Y), Z) → U1_GGA(X, Y, Z, mult_in_gga(X, Y, W))
MULT_IN_GGA(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → U2_GGA(X, Y, Z, sum_in_gga(W, X, Z))
U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → SUM_IN_GGA(W, X, Z)
SUM_IN_GGA(X, s(Y), s(Z)) → U3_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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(
x1,
x4)
U2_gga(
x1,
x2,
x3,
x4) =
U2_gga(
x4)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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(
x1,
x4)
U2_GGA(
x1,
x2,
x3,
x4) =
U2_GGA(
x4)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_IN_GGA(
x1,
x2)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_GGA(
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:
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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(
x1,
x4)
U2_gga(
x1,
x2,
x3,
x4) =
U2_gga(
x4)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_out_gga(
x3)
U3_gga(
x1,
x2,
x3,
x4) =
U3_gga(
x4)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_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:
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
SUM_IN_GGA(
x1,
x2,
x3) =
SUM_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:
SUM_IN_GGA(X, s(Y)) → SUM_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:
- SUM_IN_GGA(X, s(Y)) → SUM_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(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
The TRS R consists of the following rules:
mult_in_gga(X1, 0, 0) → mult_out_gga(X1, 0, 0)
mult_in_gga(X, s(Y), Z) → U1_gga(X, Y, Z, mult_in_gga(X, Y, W))
U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U2_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U3_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(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(
x1,
x4)
U2_gga(
x1,
x2,
x3,
x4) =
U2_gga(
x4)
sum_in_gga(
x1,
x2,
x3) =
sum_in_gga(
x1,
x2)
sum_out_gga(
x1,
x2,
x3) =
sum_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
(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(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
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(X, s(Y)) → MULT_IN_GGA(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.