(0) Obligation:
Clauses:
factor(.(X, []), X).
factor(.(X, .(Y, Xs)), T) :- ','(times(X, Y, Z), factor(.(Z, Xs), T)).
times(0, X_, 0).
times(s(X), Y, Z) :- ','(times(X, Y, XY), plus(XY, Y, Z)).
plus(0, X, X).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
Query: factor(g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
timesB(s(X1), X2, X3) :- timesB(X1, X2, X4).
timesB(s(X1), X2, X3) :- ','(timescB(X1, X2, X4), plusC(X4, X2, X3)).
plusC(s(X1), X2, s(X3)) :- plusC(X1, X2, X3).
factorA(.(0, .(X1, X2)), X3) :- factorA(.(0, X2), X3).
factorA(.(s(X1), .(X2, X3)), X4) :- timesB(X1, X2, X5).
factorA(.(s(X1), .(X2, X3)), X4) :- ','(timescB(X1, X2, X5), plusC(X5, X2, X6)).
factorA(.(s(X1), .(X2, X3)), X4) :- ','(timescB(X1, X2, X5), ','(pluscC(X5, X2, X6), factorA(.(X6, X3), X4))).
Clauses:
factorcA(.(X1, []), X1).
factorcA(.(0, .(X1, X2)), X3) :- factorcA(.(0, X2), X3).
factorcA(.(s(X1), .(X2, X3)), X4) :- ','(timescB(X1, X2, X5), ','(pluscC(X5, X2, X6), factorcA(.(X6, X3), X4))).
timescB(0, X1, 0).
timescB(s(X1), X2, X3) :- ','(timescB(X1, X2, X4), pluscC(X4, X2, X3)).
pluscC(0, X1, X1).
pluscC(s(X1), X2, s(X3)) :- pluscC(X1, X2, X3).
Afs:
factorA(x1, x2) = factorA(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
factorA_in: (b,f)
timesB_in: (b,b,f)
timescB_in: (b,b,f)
pluscC_in: (b,b,f)
plusC_in: (b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
FACTORA_IN_GA(.(0, .(X1, X2)), X3) → U5_GA(X1, X2, X3, factorA_in_ga(.(0, X2), X3))
FACTORA_IN_GA(.(0, .(X1, X2)), X3) → FACTORA_IN_GA(.(0, X2), X3)
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, timesB_in_gga(X1, X2, X5))
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → TIMESB_IN_GGA(X1, X2, X5)
TIMESB_IN_GGA(s(X1), X2, X3) → U1_GGA(X1, X2, X3, timesB_in_gga(X1, X2, X4))
TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
TIMESB_IN_GGA(s(X1), X2, X3) → U2_GGA(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, plusC_in_gga(X4, X2, X3))
U2_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → PLUSC_IN_GGA(X4, X2, X3)
PLUSC_IN_GGA(s(X1), X2, s(X3)) → U4_GGA(X1, X2, X3, plusC_in_gga(X1, X2, X3))
PLUSC_IN_GGA(s(X1), X2, s(X3)) → PLUSC_IN_GGA(X1, X2, X3)
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, timescB_in_gga(X1, X2, X5))
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → U8_GA(X1, X2, X3, X4, plusC_in_gga(X5, X2, X6))
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → PLUSC_IN_GGA(X5, X2, X6)
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → U9_GA(X1, X2, X3, X4, pluscC_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, pluscC_out_gga(X5, X2, X6)) → U10_GA(X1, X2, X3, X4, factorA_in_ga(.(X6, X3), X4))
U9_GA(X1, X2, X3, X4, pluscC_out_gga(X5, X2, X6)) → FACTORA_IN_GA(.(X6, X3), X4)
The TRS R consists of the following rules:
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U16_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U16_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, X3, pluscC_in_gga(X4, X2, X3))
pluscC_in_gga(0, X1, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2, s(X3)) → U18_gga(X1, X2, X3, pluscC_in_gga(X1, X2, X3))
U18_gga(X1, X2, X3, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, X3, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The argument filtering Pi contains the following mapping:
factorA_in_ga(
x1,
x2) =
factorA_in_ga(
x1)
.(
x1,
x2) =
.(
x1,
x2)
0 =
0
s(
x1) =
s(
x1)
timesB_in_gga(
x1,
x2,
x3) =
timesB_in_gga(
x1,
x2)
timescB_in_gga(
x1,
x2,
x3) =
timescB_in_gga(
x1,
x2)
timescB_out_gga(
x1,
x2,
x3) =
timescB_out_gga(
x1,
x2,
x3)
U16_gga(
x1,
x2,
x3,
x4) =
U16_gga(
x1,
x2,
x4)
U17_gga(
x1,
x2,
x3,
x4) =
U17_gga(
x1,
x2,
x4)
pluscC_in_gga(
x1,
x2,
x3) =
pluscC_in_gga(
x1,
x2)
pluscC_out_gga(
x1,
x2,
x3) =
pluscC_out_gga(
x1,
x2,
x3)
U18_gga(
x1,
x2,
x3,
x4) =
U18_gga(
x1,
x2,
x4)
plusC_in_gga(
x1,
x2,
x3) =
plusC_in_gga(
x1,
x2)
FACTORA_IN_GA(
x1,
x2) =
FACTORA_IN_GA(
x1)
U5_GA(
x1,
x2,
x3,
x4) =
U5_GA(
x1,
x2,
x4)
U6_GA(
x1,
x2,
x3,
x4,
x5) =
U6_GA(
x1,
x2,
x3,
x5)
TIMESB_IN_GGA(
x1,
x2,
x3) =
TIMESB_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)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_GGA(
x1,
x2,
x4)
PLUSC_IN_GGA(
x1,
x2,
x3) =
PLUSC_IN_GGA(
x1,
x2)
U4_GGA(
x1,
x2,
x3,
x4) =
U4_GGA(
x1,
x2,
x4)
U7_GA(
x1,
x2,
x3,
x4,
x5) =
U7_GA(
x1,
x2,
x3,
x5)
U8_GA(
x1,
x2,
x3,
x4,
x5) =
U8_GA(
x1,
x2,
x3,
x5)
U9_GA(
x1,
x2,
x3,
x4,
x5) =
U9_GA(
x1,
x2,
x3,
x5)
U10_GA(
x1,
x2,
x3,
x4,
x5) =
U10_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORA_IN_GA(.(0, .(X1, X2)), X3) → U5_GA(X1, X2, X3, factorA_in_ga(.(0, X2), X3))
FACTORA_IN_GA(.(0, .(X1, X2)), X3) → FACTORA_IN_GA(.(0, X2), X3)
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, timesB_in_gga(X1, X2, X5))
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → TIMESB_IN_GGA(X1, X2, X5)
TIMESB_IN_GGA(s(X1), X2, X3) → U1_GGA(X1, X2, X3, timesB_in_gga(X1, X2, X4))
TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
TIMESB_IN_GGA(s(X1), X2, X3) → U2_GGA(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, plusC_in_gga(X4, X2, X3))
U2_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → PLUSC_IN_GGA(X4, X2, X3)
PLUSC_IN_GGA(s(X1), X2, s(X3)) → U4_GGA(X1, X2, X3, plusC_in_gga(X1, X2, X3))
PLUSC_IN_GGA(s(X1), X2, s(X3)) → PLUSC_IN_GGA(X1, X2, X3)
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, timescB_in_gga(X1, X2, X5))
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → U8_GA(X1, X2, X3, X4, plusC_in_gga(X5, X2, X6))
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → PLUSC_IN_GGA(X5, X2, X6)
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → U9_GA(X1, X2, X3, X4, pluscC_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, pluscC_out_gga(X5, X2, X6)) → U10_GA(X1, X2, X3, X4, factorA_in_ga(.(X6, X3), X4))
U9_GA(X1, X2, X3, X4, pluscC_out_gga(X5, X2, X6)) → FACTORA_IN_GA(.(X6, X3), X4)
The TRS R consists of the following rules:
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U16_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U16_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, X3, pluscC_in_gga(X4, X2, X3))
pluscC_in_gga(0, X1, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2, s(X3)) → U18_gga(X1, X2, X3, pluscC_in_gga(X1, X2, X3))
U18_gga(X1, X2, X3, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, X3, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The argument filtering Pi contains the following mapping:
factorA_in_ga(
x1,
x2) =
factorA_in_ga(
x1)
.(
x1,
x2) =
.(
x1,
x2)
0 =
0
s(
x1) =
s(
x1)
timesB_in_gga(
x1,
x2,
x3) =
timesB_in_gga(
x1,
x2)
timescB_in_gga(
x1,
x2,
x3) =
timescB_in_gga(
x1,
x2)
timescB_out_gga(
x1,
x2,
x3) =
timescB_out_gga(
x1,
x2,
x3)
U16_gga(
x1,
x2,
x3,
x4) =
U16_gga(
x1,
x2,
x4)
U17_gga(
x1,
x2,
x3,
x4) =
U17_gga(
x1,
x2,
x4)
pluscC_in_gga(
x1,
x2,
x3) =
pluscC_in_gga(
x1,
x2)
pluscC_out_gga(
x1,
x2,
x3) =
pluscC_out_gga(
x1,
x2,
x3)
U18_gga(
x1,
x2,
x3,
x4) =
U18_gga(
x1,
x2,
x4)
plusC_in_gga(
x1,
x2,
x3) =
plusC_in_gga(
x1,
x2)
FACTORA_IN_GA(
x1,
x2) =
FACTORA_IN_GA(
x1)
U5_GA(
x1,
x2,
x3,
x4) =
U5_GA(
x1,
x2,
x4)
U6_GA(
x1,
x2,
x3,
x4,
x5) =
U6_GA(
x1,
x2,
x3,
x5)
TIMESB_IN_GGA(
x1,
x2,
x3) =
TIMESB_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)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_GGA(
x1,
x2,
x4)
PLUSC_IN_GGA(
x1,
x2,
x3) =
PLUSC_IN_GGA(
x1,
x2)
U4_GGA(
x1,
x2,
x3,
x4) =
U4_GGA(
x1,
x2,
x4)
U7_GA(
x1,
x2,
x3,
x4,
x5) =
U7_GA(
x1,
x2,
x3,
x5)
U8_GA(
x1,
x2,
x3,
x4,
x5) =
U8_GA(
x1,
x2,
x3,
x5)
U9_GA(
x1,
x2,
x3,
x4,
x5) =
U9_GA(
x1,
x2,
x3,
x5)
U10_GA(
x1,
x2,
x3,
x4,
x5) =
U10_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 11 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PLUSC_IN_GGA(s(X1), X2, s(X3)) → PLUSC_IN_GGA(X1, X2, X3)
The TRS R consists of the following rules:
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U16_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U16_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, X3, pluscC_in_gga(X4, X2, X3))
pluscC_in_gga(0, X1, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2, s(X3)) → U18_gga(X1, X2, X3, pluscC_in_gga(X1, X2, X3))
U18_gga(X1, X2, X3, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, X3, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The argument filtering Pi contains the following mapping:
0 =
0
s(
x1) =
s(
x1)
timescB_in_gga(
x1,
x2,
x3) =
timescB_in_gga(
x1,
x2)
timescB_out_gga(
x1,
x2,
x3) =
timescB_out_gga(
x1,
x2,
x3)
U16_gga(
x1,
x2,
x3,
x4) =
U16_gga(
x1,
x2,
x4)
U17_gga(
x1,
x2,
x3,
x4) =
U17_gga(
x1,
x2,
x4)
pluscC_in_gga(
x1,
x2,
x3) =
pluscC_in_gga(
x1,
x2)
pluscC_out_gga(
x1,
x2,
x3) =
pluscC_out_gga(
x1,
x2,
x3)
U18_gga(
x1,
x2,
x3,
x4) =
U18_gga(
x1,
x2,
x4)
PLUSC_IN_GGA(
x1,
x2,
x3) =
PLUSC_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:
PLUSC_IN_GGA(s(X1), X2, s(X3)) → PLUSC_IN_GGA(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
PLUSC_IN_GGA(
x1,
x2,
x3) =
PLUSC_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:
PLUSC_IN_GGA(s(X1), X2) → PLUSC_IN_GGA(X1, X2)
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:
- PLUSC_IN_GGA(s(X1), X2) → PLUSC_IN_GGA(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
(13) YES
(14) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
The TRS R consists of the following rules:
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U16_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U16_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, X3, pluscC_in_gga(X4, X2, X3))
pluscC_in_gga(0, X1, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2, s(X3)) → U18_gga(X1, X2, X3, pluscC_in_gga(X1, X2, X3))
U18_gga(X1, X2, X3, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, X3, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The argument filtering Pi contains the following mapping:
0 =
0
s(
x1) =
s(
x1)
timescB_in_gga(
x1,
x2,
x3) =
timescB_in_gga(
x1,
x2)
timescB_out_gga(
x1,
x2,
x3) =
timescB_out_gga(
x1,
x2,
x3)
U16_gga(
x1,
x2,
x3,
x4) =
U16_gga(
x1,
x2,
x4)
U17_gga(
x1,
x2,
x3,
x4) =
U17_gga(
x1,
x2,
x4)
pluscC_in_gga(
x1,
x2,
x3) =
pluscC_in_gga(
x1,
x2)
pluscC_out_gga(
x1,
x2,
x3) =
pluscC_out_gga(
x1,
x2,
x3)
U18_gga(
x1,
x2,
x3,
x4) =
U18_gga(
x1,
x2,
x4)
TIMESB_IN_GGA(
x1,
x2,
x3) =
TIMESB_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:
TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
TIMESB_IN_GGA(
x1,
x2,
x3) =
TIMESB_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:
TIMESB_IN_GGA(s(X1), X2) → TIMESB_IN_GGA(X1, X2)
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:
- TIMESB_IN_GGA(s(X1), X2) → TIMESB_IN_GGA(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
(20) YES
(21) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORA_IN_GA(.(0, .(X1, X2)), X3) → FACTORA_IN_GA(.(0, X2), X3)
The TRS R consists of the following rules:
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U16_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U16_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, X3, pluscC_in_gga(X4, X2, X3))
pluscC_in_gga(0, X1, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2, s(X3)) → U18_gga(X1, X2, X3, pluscC_in_gga(X1, X2, X3))
U18_gga(X1, X2, X3, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, X3, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
0 =
0
s(
x1) =
s(
x1)
timescB_in_gga(
x1,
x2,
x3) =
timescB_in_gga(
x1,
x2)
timescB_out_gga(
x1,
x2,
x3) =
timescB_out_gga(
x1,
x2,
x3)
U16_gga(
x1,
x2,
x3,
x4) =
U16_gga(
x1,
x2,
x4)
U17_gga(
x1,
x2,
x3,
x4) =
U17_gga(
x1,
x2,
x4)
pluscC_in_gga(
x1,
x2,
x3) =
pluscC_in_gga(
x1,
x2)
pluscC_out_gga(
x1,
x2,
x3) =
pluscC_out_gga(
x1,
x2,
x3)
U18_gga(
x1,
x2,
x3,
x4) =
U18_gga(
x1,
x2,
x4)
FACTORA_IN_GA(
x1,
x2) =
FACTORA_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:
FACTORA_IN_GA(.(0, .(X1, X2)), X3) → FACTORA_IN_GA(.(0, X2), X3)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
0 =
0
FACTORA_IN_GA(
x1,
x2) =
FACTORA_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:
FACTORA_IN_GA(.(0, .(X1, X2))) → FACTORA_IN_GA(.(0, X2))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(26) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
FACTORA_IN_GA(.(0, .(X1, X2))) → FACTORA_IN_GA(.(0, X2))
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 1 + x1 + 2·x2
POL(0) = 0
POL(FACTORA_IN_GA(x1)) = 2·x1
(27) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(28) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(29) YES
(30) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORA_IN_GA(.(s(X1), .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, timescB_in_gga(X1, X2, X5))
U7_GA(X1, X2, X3, X4, timescB_out_gga(X1, X2, X5)) → U9_GA(X1, X2, X3, X4, pluscC_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, pluscC_out_gga(X5, X2, X6)) → FACTORA_IN_GA(.(X6, X3), X4)
The TRS R consists of the following rules:
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U16_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U16_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, X3, pluscC_in_gga(X4, X2, X3))
pluscC_in_gga(0, X1, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2, s(X3)) → U18_gga(X1, X2, X3, pluscC_in_gga(X1, X2, X3))
U18_gga(X1, X2, X3, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, X3, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
0 =
0
s(
x1) =
s(
x1)
timescB_in_gga(
x1,
x2,
x3) =
timescB_in_gga(
x1,
x2)
timescB_out_gga(
x1,
x2,
x3) =
timescB_out_gga(
x1,
x2,
x3)
U16_gga(
x1,
x2,
x3,
x4) =
U16_gga(
x1,
x2,
x4)
U17_gga(
x1,
x2,
x3,
x4) =
U17_gga(
x1,
x2,
x4)
pluscC_in_gga(
x1,
x2,
x3) =
pluscC_in_gga(
x1,
x2)
pluscC_out_gga(
x1,
x2,
x3) =
pluscC_out_gga(
x1,
x2,
x3)
U18_gga(
x1,
x2,
x3,
x4) =
U18_gga(
x1,
x2,
x4)
FACTORA_IN_GA(
x1,
x2) =
FACTORA_IN_GA(
x1)
U7_GA(
x1,
x2,
x3,
x4,
x5) =
U7_GA(
x1,
x2,
x3,
x5)
U9_GA(
x1,
x2,
x3,
x4,
x5) =
U9_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(31) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(32) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FACTORA_IN_GA(.(s(X1), .(X2, X3))) → U7_GA(X1, X2, X3, timescB_in_gga(X1, X2))
U7_GA(X1, X2, X3, timescB_out_gga(X1, X2, X5)) → U9_GA(X1, X2, X3, pluscC_in_gga(X5, X2))
U9_GA(X1, X2, X3, pluscC_out_gga(X5, X2, X6)) → FACTORA_IN_GA(.(X6, X3))
The TRS R consists of the following rules:
timescB_in_gga(0, X1) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2) → U16_gga(X1, X2, timescB_in_gga(X1, X2))
U16_gga(X1, X2, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, pluscC_in_gga(X4, X2))
pluscC_in_gga(0, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2) → U18_gga(X1, X2, pluscC_in_gga(X1, X2))
U18_gga(X1, X2, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The set Q consists of the following terms:
timescB_in_gga(x0, x1)
U16_gga(x0, x1, x2)
pluscC_in_gga(x0, x1)
U18_gga(x0, x1, x2)
U17_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(33) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
FACTORA_IN_GA(.(s(X1), .(X2, X3))) → U7_GA(X1, X2, X3, timescB_in_gga(X1, X2))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 1 + x2
POL(0) = 0
POL(FACTORA_IN_GA(x1)) = x1
POL(U16_gga(x1, x2, x3)) = 0
POL(U17_gga(x1, x2, x3)) = 0
POL(U18_gga(x1, x2, x3)) = 1
POL(U7_GA(x1, x2, x3, x4)) = 1 + x3
POL(U9_GA(x1, x2, x3, x4)) = x3 + x4
POL(pluscC_in_gga(x1, x2)) = 1
POL(pluscC_out_gga(x1, x2, x3)) = 1
POL(s(x1)) = 0
POL(timescB_in_gga(x1, x2)) = 0
POL(timescB_out_gga(x1, x2, x3)) = 0
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
pluscC_in_gga(0, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2) → U18_gga(X1, X2, pluscC_in_gga(X1, X2))
U18_gga(X1, X2, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
(34) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U7_GA(X1, X2, X3, timescB_out_gga(X1, X2, X5)) → U9_GA(X1, X2, X3, pluscC_in_gga(X5, X2))
U9_GA(X1, X2, X3, pluscC_out_gga(X5, X2, X6)) → FACTORA_IN_GA(.(X6, X3))
The TRS R consists of the following rules:
timescB_in_gga(0, X1) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2) → U16_gga(X1, X2, timescB_in_gga(X1, X2))
U16_gga(X1, X2, timescB_out_gga(X1, X2, X4)) → U17_gga(X1, X2, pluscC_in_gga(X4, X2))
pluscC_in_gga(0, X1) → pluscC_out_gga(0, X1, X1)
pluscC_in_gga(s(X1), X2) → U18_gga(X1, X2, pluscC_in_gga(X1, X2))
U18_gga(X1, X2, pluscC_out_gga(X1, X2, X3)) → pluscC_out_gga(s(X1), X2, s(X3))
U17_gga(X1, X2, pluscC_out_gga(X4, X2, X3)) → timescB_out_gga(s(X1), X2, X3)
The set Q consists of the following terms:
timescB_in_gga(x0, x1)
U16_gga(x0, x1, x2)
pluscC_in_gga(x0, x1)
U18_gga(x0, x1, x2)
U17_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(35) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(36) TRUE