(0) Obligation:
Clauses:
times(X, Y, Z) :- mult(X, Y, 0, Z).
mult(0, Y, 0, 0).
mult(s(U), Y, 0, Z) :- mult(U, Y, Y, Z).
mult(X, Y, s(W), s(Z)) :- mult(X, Y, W, Z).
Queries:
times(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:
times_in: (b,b,f)
mult_in: (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_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
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:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
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:
TIMES_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
TIMES_IN_GGA(X, Y, Z) → MULT_IN_GGGA(X, Y, 0, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
MULT_IN_GGGA(X, Y, s(W), s(Z)) → U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x3)
TIMES_IN_GGA(
x1,
x2,
x3) =
TIMES_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4) =
U1_GGA(
x4)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
U2_GGGA(
x1,
x2,
x3,
x4) =
U2_GGGA(
x4)
U3_GGGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGGA(
x5)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
TIMES_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
TIMES_IN_GGA(X, Y, Z) → MULT_IN_GGGA(X, Y, 0, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
MULT_IN_GGGA(X, Y, s(W), s(Z)) → U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x3)
TIMES_IN_GGA(
x1,
x2,
x3) =
TIMES_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4) =
U1_GGA(
x4)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
U2_GGGA(
x1,
x2,
x3,
x4) =
U2_GGGA(
x4)
U3_GGGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGGA(
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x3)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
We have to consider all (P,R,Pi)-chains
(7) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
0 =
0
s(
x1) =
s(
x1)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
We have to consider all (P,R,Pi)-chains
(9) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MULT_IN_GGGA(X, Y, s(W)) → MULT_IN_GGGA(X, Y, W)
MULT_IN_GGGA(s(U), Y, 0) → MULT_IN_GGGA(U, Y, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) 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_GGGA(X, Y, s(W)) → MULT_IN_GGGA(X, Y, W)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
- MULT_IN_GGGA(s(U), Y, 0) → MULT_IN_GGGA(U, Y, Y)
The graph contains the following edges 1 > 1, 2 >= 2, 2 >= 3
(12) TRUE
(13) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
times_in: (b,b,f)
mult_in: (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_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x1,
x2,
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x1,
x2,
x3,
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x1,
x2,
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x1,
x2,
x3,
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x1,
x2,
x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(14) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x1,
x2,
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x1,
x2,
x3,
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x1,
x2,
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x1,
x2,
x3,
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x1,
x2,
x3)
(15) 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:
TIMES_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
TIMES_IN_GGA(X, Y, Z) → MULT_IN_GGGA(X, Y, 0, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
MULT_IN_GGGA(X, Y, s(W), s(Z)) → U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x1,
x2,
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x1,
x2,
x3,
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x1,
x2,
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x1,
x2,
x3,
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x1,
x2,
x3)
TIMES_IN_GGA(
x1,
x2,
x3) =
TIMES_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4) =
U1_GGA(
x1,
x2,
x4)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
U2_GGGA(
x1,
x2,
x3,
x4) =
U2_GGGA(
x1,
x2,
x4)
U3_GGGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(16) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
TIMES_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
TIMES_IN_GGA(X, Y, Z) → MULT_IN_GGGA(X, Y, 0, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
MULT_IN_GGGA(X, Y, s(W), s(Z)) → U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x1,
x2,
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x1,
x2,
x3,
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x1,
x2,
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x1,
x2,
x3,
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x1,
x2,
x3)
TIMES_IN_GGA(
x1,
x2,
x3) =
TIMES_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4) =
U1_GGA(
x1,
x2,
x4)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
U2_GGGA(
x1,
x2,
x3,
x4) =
U2_GGGA(
x1,
x2,
x4)
U3_GGGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(17) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.
(18) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
The TRS R consists of the following rules:
times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)
The argument filtering Pi contains the following mapping:
times_in_gga(
x1,
x2,
x3) =
times_in_gga(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4) =
U1_gga(
x1,
x2,
x4)
mult_in_ggga(
x1,
x2,
x3,
x4) =
mult_in_ggga(
x1,
x2,
x3)
0 =
0
mult_out_ggga(
x1,
x2,
x3,
x4) =
mult_out_ggga(
x1,
x2,
x3,
x4)
s(
x1) =
s(
x1)
U2_ggga(
x1,
x2,
x3,
x4) =
U2_ggga(
x1,
x2,
x4)
U3_ggga(
x1,
x2,
x3,
x4,
x5) =
U3_ggga(
x1,
x2,
x3,
x5)
times_out_gga(
x1,
x2,
x3) =
times_out_gga(
x1,
x2,
x3)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
We have to consider all (P,R,Pi)-chains
(19) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(20) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
0 =
0
s(
x1) =
s(
x1)
MULT_IN_GGGA(
x1,
x2,
x3,
x4) =
MULT_IN_GGGA(
x1,
x2,
x3)
We have to consider all (P,R,Pi)-chains
(21) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MULT_IN_GGGA(X, Y, s(W)) → MULT_IN_GGGA(X, Y, W)
MULT_IN_GGGA(s(U), Y, 0) → MULT_IN_GGGA(U, Y, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.