Left Termination of the query pattern log2(b,f) w.r.t. the given Prolog program could successfully be proven:



↳ PROLOG
  ↳ PrologToPiTRSProof

log22(X, Y) :- log24(X, 00, s1(00), Y).
log24(s12 (X), Half, Acc, Y) :- log24(X, s1(Half), Acc, Y).
log24(X, s12 (Half), Acc, Y) :- small1(X), log24(Half, s1(00), s1(Acc), Y).
log24(X, Half, Y, Y) :- small1(X), small1(Half).
small1(00).
small1(s1(00)).


With regard to the inferred argument filtering the predicates were used in the following modes:
log22: (b,f)
log24: (b,b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


log2_2_in_ga2(X, Y) -> if_log2_2_in_1_ga3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
log2_4_in_ggga4(s_11(s_11(X)), Half, Acc, Y) -> if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
log2_4_in_ggga4(X, s_11(s_11(Half)), Acc, Y) -> if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_in_g1(X))
small_1_in_g1(0_0) -> small_1_out_g1(0_0)
small_1_in_g1(s_11(0_0)) -> small_1_out_g1(s_11(0_0))
if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_out_g1(X)) -> if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
log2_4_in_ggga4(X, Half, Y, Y) -> if_log2_4_in_4_ggga4(X, Half, Y, small_1_in_g1(X))
if_log2_4_in_4_ggga4(X, Half, Y, small_1_out_g1(X)) -> if_log2_4_in_5_ggga4(X, Half, Y, small_1_in_g1(Half))
if_log2_4_in_5_ggga4(X, Half, Y, small_1_out_g1(Half)) -> log2_4_out_ggga4(X, Half, Y, Y)
if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(Half, s_11(0_0), s_11(Acc), Y)) -> log2_4_out_ggga4(X, s_11(s_11(Half)), Acc, Y)
if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(X, s_11(Half), Acc, Y)) -> log2_4_out_ggga4(s_11(s_11(X)), Half, Acc, Y)
if_log2_2_in_1_ga3(X, Y, log2_4_out_ggga4(X, 0_0, s_11(0_0), Y)) -> log2_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ga2(x1, x2)  =  log2_2_in_ga1(x1)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_log2_2_in_1_ga3(x1, x2, x3)  =  if_log2_2_in_1_ga1(x3)
log2_4_in_ggga4(x1, x2, x3, x4)  =  log2_4_in_ggga3(x1, x2, x3)
if_log2_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_1_ggga1(x5)
if_log2_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_2_ggga3(x2, x3, x5)
small_1_in_g1(x1)  =  small_1_in_g1(x1)
small_1_out_g1(x1)  =  small_1_out_g
if_log2_4_in_3_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_3_ggga1(x5)
if_log2_4_in_4_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_4_ggga3(x2, x3, x4)
if_log2_4_in_5_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_5_ggga2(x3, x4)
log2_4_out_ggga4(x1, x2, x3, x4)  =  log2_4_out_ggga1(x4)
log2_2_out_ga2(x1, x2)  =  log2_2_out_ga1(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

log2_2_in_ga2(X, Y) -> if_log2_2_in_1_ga3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
log2_4_in_ggga4(s_11(s_11(X)), Half, Acc, Y) -> if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
log2_4_in_ggga4(X, s_11(s_11(Half)), Acc, Y) -> if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_in_g1(X))
small_1_in_g1(0_0) -> small_1_out_g1(0_0)
small_1_in_g1(s_11(0_0)) -> small_1_out_g1(s_11(0_0))
if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_out_g1(X)) -> if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
log2_4_in_ggga4(X, Half, Y, Y) -> if_log2_4_in_4_ggga4(X, Half, Y, small_1_in_g1(X))
if_log2_4_in_4_ggga4(X, Half, Y, small_1_out_g1(X)) -> if_log2_4_in_5_ggga4(X, Half, Y, small_1_in_g1(Half))
if_log2_4_in_5_ggga4(X, Half, Y, small_1_out_g1(Half)) -> log2_4_out_ggga4(X, Half, Y, Y)
if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(Half, s_11(0_0), s_11(Acc), Y)) -> log2_4_out_ggga4(X, s_11(s_11(Half)), Acc, Y)
if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(X, s_11(Half), Acc, Y)) -> log2_4_out_ggga4(s_11(s_11(X)), Half, Acc, Y)
if_log2_2_in_1_ga3(X, Y, log2_4_out_ggga4(X, 0_0, s_11(0_0), Y)) -> log2_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ga2(x1, x2)  =  log2_2_in_ga1(x1)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_log2_2_in_1_ga3(x1, x2, x3)  =  if_log2_2_in_1_ga1(x3)
log2_4_in_ggga4(x1, x2, x3, x4)  =  log2_4_in_ggga3(x1, x2, x3)
if_log2_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_1_ggga1(x5)
if_log2_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_2_ggga3(x2, x3, x5)
small_1_in_g1(x1)  =  small_1_in_g1(x1)
small_1_out_g1(x1)  =  small_1_out_g
if_log2_4_in_3_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_3_ggga1(x5)
if_log2_4_in_4_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_4_ggga3(x2, x3, x4)
if_log2_4_in_5_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_5_ggga2(x3, x4)
log2_4_out_ggga4(x1, x2, x3, x4)  =  log2_4_out_ggga1(x4)
log2_2_out_ga2(x1, x2)  =  log2_2_out_ga1(x2)


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

LOG2_2_IN_GA2(X, Y) -> IF_LOG2_2_IN_1_GA3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
LOG2_2_IN_GA2(X, Y) -> LOG2_4_IN_GGGA4(X, 0_0, s_11(0_0), Y)
LOG2_4_IN_GGGA4(s_11(s_11(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_GGGA5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
LOG2_4_IN_GGGA4(s_11(s_11(X)), Half, Acc, Y) -> LOG2_4_IN_GGGA4(X, s_11(Half), Acc, Y)
LOG2_4_IN_GGGA4(X, s_11(s_11(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_in_g1(X))
LOG2_4_IN_GGGA4(X, s_11(s_11(Half)), Acc, Y) -> SMALL_1_IN_G1(X)
IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_out_g1(X)) -> IF_LOG2_4_IN_3_GGGA5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_out_g1(X)) -> LOG2_4_IN_GGGA4(Half, s_11(0_0), s_11(Acc), Y)
LOG2_4_IN_GGGA4(X, Half, Y, Y) -> IF_LOG2_4_IN_4_GGGA4(X, Half, Y, small_1_in_g1(X))
LOG2_4_IN_GGGA4(X, Half, Y, Y) -> SMALL_1_IN_G1(X)
IF_LOG2_4_IN_4_GGGA4(X, Half, Y, small_1_out_g1(X)) -> IF_LOG2_4_IN_5_GGGA4(X, Half, Y, small_1_in_g1(Half))
IF_LOG2_4_IN_4_GGGA4(X, Half, Y, small_1_out_g1(X)) -> SMALL_1_IN_G1(Half)

The TRS R consists of the following rules:

log2_2_in_ga2(X, Y) -> if_log2_2_in_1_ga3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
log2_4_in_ggga4(s_11(s_11(X)), Half, Acc, Y) -> if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
log2_4_in_ggga4(X, s_11(s_11(Half)), Acc, Y) -> if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_in_g1(X))
small_1_in_g1(0_0) -> small_1_out_g1(0_0)
small_1_in_g1(s_11(0_0)) -> small_1_out_g1(s_11(0_0))
if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_out_g1(X)) -> if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
log2_4_in_ggga4(X, Half, Y, Y) -> if_log2_4_in_4_ggga4(X, Half, Y, small_1_in_g1(X))
if_log2_4_in_4_ggga4(X, Half, Y, small_1_out_g1(X)) -> if_log2_4_in_5_ggga4(X, Half, Y, small_1_in_g1(Half))
if_log2_4_in_5_ggga4(X, Half, Y, small_1_out_g1(Half)) -> log2_4_out_ggga4(X, Half, Y, Y)
if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(Half, s_11(0_0), s_11(Acc), Y)) -> log2_4_out_ggga4(X, s_11(s_11(Half)), Acc, Y)
if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(X, s_11(Half), Acc, Y)) -> log2_4_out_ggga4(s_11(s_11(X)), Half, Acc, Y)
if_log2_2_in_1_ga3(X, Y, log2_4_out_ggga4(X, 0_0, s_11(0_0), Y)) -> log2_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ga2(x1, x2)  =  log2_2_in_ga1(x1)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_log2_2_in_1_ga3(x1, x2, x3)  =  if_log2_2_in_1_ga1(x3)
log2_4_in_ggga4(x1, x2, x3, x4)  =  log2_4_in_ggga3(x1, x2, x3)
if_log2_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_1_ggga1(x5)
if_log2_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_2_ggga3(x2, x3, x5)
small_1_in_g1(x1)  =  small_1_in_g1(x1)
small_1_out_g1(x1)  =  small_1_out_g
if_log2_4_in_3_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_3_ggga1(x5)
if_log2_4_in_4_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_4_ggga3(x2, x3, x4)
if_log2_4_in_5_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_5_ggga2(x3, x4)
log2_4_out_ggga4(x1, x2, x3, x4)  =  log2_4_out_ggga1(x4)
log2_2_out_ga2(x1, x2)  =  log2_2_out_ga1(x2)
IF_LOG2_4_IN_3_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_3_GGGA1(x5)
SMALL_1_IN_G1(x1)  =  SMALL_1_IN_G1(x1)
IF_LOG2_2_IN_1_GA3(x1, x2, x3)  =  IF_LOG2_2_IN_1_GA1(x3)
IF_LOG2_4_IN_5_GGGA4(x1, x2, x3, x4)  =  IF_LOG2_4_IN_5_GGGA2(x3, x4)
IF_LOG2_4_IN_1_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_1_GGGA1(x5)
LOG2_4_IN_GGGA4(x1, x2, x3, x4)  =  LOG2_4_IN_GGGA3(x1, x2, x3)
LOG2_2_IN_GA2(x1, x2)  =  LOG2_2_IN_GA1(x1)
IF_LOG2_4_IN_4_GGGA4(x1, x2, x3, x4)  =  IF_LOG2_4_IN_4_GGGA3(x2, x3, x4)
IF_LOG2_4_IN_2_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_2_GGGA3(x2, x3, x5)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

LOG2_2_IN_GA2(X, Y) -> IF_LOG2_2_IN_1_GA3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
LOG2_2_IN_GA2(X, Y) -> LOG2_4_IN_GGGA4(X, 0_0, s_11(0_0), Y)
LOG2_4_IN_GGGA4(s_11(s_11(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_GGGA5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
LOG2_4_IN_GGGA4(s_11(s_11(X)), Half, Acc, Y) -> LOG2_4_IN_GGGA4(X, s_11(Half), Acc, Y)
LOG2_4_IN_GGGA4(X, s_11(s_11(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_in_g1(X))
LOG2_4_IN_GGGA4(X, s_11(s_11(Half)), Acc, Y) -> SMALL_1_IN_G1(X)
IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_out_g1(X)) -> IF_LOG2_4_IN_3_GGGA5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_out_g1(X)) -> LOG2_4_IN_GGGA4(Half, s_11(0_0), s_11(Acc), Y)
LOG2_4_IN_GGGA4(X, Half, Y, Y) -> IF_LOG2_4_IN_4_GGGA4(X, Half, Y, small_1_in_g1(X))
LOG2_4_IN_GGGA4(X, Half, Y, Y) -> SMALL_1_IN_G1(X)
IF_LOG2_4_IN_4_GGGA4(X, Half, Y, small_1_out_g1(X)) -> IF_LOG2_4_IN_5_GGGA4(X, Half, Y, small_1_in_g1(Half))
IF_LOG2_4_IN_4_GGGA4(X, Half, Y, small_1_out_g1(X)) -> SMALL_1_IN_G1(Half)

The TRS R consists of the following rules:

log2_2_in_ga2(X, Y) -> if_log2_2_in_1_ga3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
log2_4_in_ggga4(s_11(s_11(X)), Half, Acc, Y) -> if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
log2_4_in_ggga4(X, s_11(s_11(Half)), Acc, Y) -> if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_in_g1(X))
small_1_in_g1(0_0) -> small_1_out_g1(0_0)
small_1_in_g1(s_11(0_0)) -> small_1_out_g1(s_11(0_0))
if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_out_g1(X)) -> if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
log2_4_in_ggga4(X, Half, Y, Y) -> if_log2_4_in_4_ggga4(X, Half, Y, small_1_in_g1(X))
if_log2_4_in_4_ggga4(X, Half, Y, small_1_out_g1(X)) -> if_log2_4_in_5_ggga4(X, Half, Y, small_1_in_g1(Half))
if_log2_4_in_5_ggga4(X, Half, Y, small_1_out_g1(Half)) -> log2_4_out_ggga4(X, Half, Y, Y)
if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(Half, s_11(0_0), s_11(Acc), Y)) -> log2_4_out_ggga4(X, s_11(s_11(Half)), Acc, Y)
if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(X, s_11(Half), Acc, Y)) -> log2_4_out_ggga4(s_11(s_11(X)), Half, Acc, Y)
if_log2_2_in_1_ga3(X, Y, log2_4_out_ggga4(X, 0_0, s_11(0_0), Y)) -> log2_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ga2(x1, x2)  =  log2_2_in_ga1(x1)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_log2_2_in_1_ga3(x1, x2, x3)  =  if_log2_2_in_1_ga1(x3)
log2_4_in_ggga4(x1, x2, x3, x4)  =  log2_4_in_ggga3(x1, x2, x3)
if_log2_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_1_ggga1(x5)
if_log2_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_2_ggga3(x2, x3, x5)
small_1_in_g1(x1)  =  small_1_in_g1(x1)
small_1_out_g1(x1)  =  small_1_out_g
if_log2_4_in_3_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_3_ggga1(x5)
if_log2_4_in_4_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_4_ggga3(x2, x3, x4)
if_log2_4_in_5_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_5_ggga2(x3, x4)
log2_4_out_ggga4(x1, x2, x3, x4)  =  log2_4_out_ggga1(x4)
log2_2_out_ga2(x1, x2)  =  log2_2_out_ga1(x2)
IF_LOG2_4_IN_3_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_3_GGGA1(x5)
SMALL_1_IN_G1(x1)  =  SMALL_1_IN_G1(x1)
IF_LOG2_2_IN_1_GA3(x1, x2, x3)  =  IF_LOG2_2_IN_1_GA1(x3)
IF_LOG2_4_IN_5_GGGA4(x1, x2, x3, x4)  =  IF_LOG2_4_IN_5_GGGA2(x3, x4)
IF_LOG2_4_IN_1_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_1_GGGA1(x5)
LOG2_4_IN_GGGA4(x1, x2, x3, x4)  =  LOG2_4_IN_GGGA3(x1, x2, x3)
LOG2_2_IN_GA2(x1, x2)  =  LOG2_2_IN_GA1(x1)
IF_LOG2_4_IN_4_GGGA4(x1, x2, x3, x4)  =  IF_LOG2_4_IN_4_GGGA3(x2, x3, x4)
IF_LOG2_4_IN_2_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_2_GGGA3(x2, x3, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 1 SCC with 9 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_out_g1(X)) -> LOG2_4_IN_GGGA4(Half, s_11(0_0), s_11(Acc), Y)
LOG2_4_IN_GGGA4(X, s_11(s_11(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_in_g1(X))
LOG2_4_IN_GGGA4(s_11(s_11(X)), Half, Acc, Y) -> LOG2_4_IN_GGGA4(X, s_11(Half), Acc, Y)

The TRS R consists of the following rules:

log2_2_in_ga2(X, Y) -> if_log2_2_in_1_ga3(X, Y, log2_4_in_ggga4(X, 0_0, s_11(0_0), Y))
log2_4_in_ggga4(s_11(s_11(X)), Half, Acc, Y) -> if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(X, s_11(Half), Acc, Y))
log2_4_in_ggga4(X, s_11(s_11(Half)), Acc, Y) -> if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_in_g1(X))
small_1_in_g1(0_0) -> small_1_out_g1(0_0)
small_1_in_g1(s_11(0_0)) -> small_1_out_g1(s_11(0_0))
if_log2_4_in_2_ggga5(X, Half, Acc, Y, small_1_out_g1(X)) -> if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_in_ggga4(Half, s_11(0_0), s_11(Acc), Y))
log2_4_in_ggga4(X, Half, Y, Y) -> if_log2_4_in_4_ggga4(X, Half, Y, small_1_in_g1(X))
if_log2_4_in_4_ggga4(X, Half, Y, small_1_out_g1(X)) -> if_log2_4_in_5_ggga4(X, Half, Y, small_1_in_g1(Half))
if_log2_4_in_5_ggga4(X, Half, Y, small_1_out_g1(Half)) -> log2_4_out_ggga4(X, Half, Y, Y)
if_log2_4_in_3_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(Half, s_11(0_0), s_11(Acc), Y)) -> log2_4_out_ggga4(X, s_11(s_11(Half)), Acc, Y)
if_log2_4_in_1_ggga5(X, Half, Acc, Y, log2_4_out_ggga4(X, s_11(Half), Acc, Y)) -> log2_4_out_ggga4(s_11(s_11(X)), Half, Acc, Y)
if_log2_2_in_1_ga3(X, Y, log2_4_out_ggga4(X, 0_0, s_11(0_0), Y)) -> log2_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ga2(x1, x2)  =  log2_2_in_ga1(x1)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_log2_2_in_1_ga3(x1, x2, x3)  =  if_log2_2_in_1_ga1(x3)
log2_4_in_ggga4(x1, x2, x3, x4)  =  log2_4_in_ggga3(x1, x2, x3)
if_log2_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_1_ggga1(x5)
if_log2_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_2_ggga3(x2, x3, x5)
small_1_in_g1(x1)  =  small_1_in_g1(x1)
small_1_out_g1(x1)  =  small_1_out_g
if_log2_4_in_3_ggga5(x1, x2, x3, x4, x5)  =  if_log2_4_in_3_ggga1(x5)
if_log2_4_in_4_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_4_ggga3(x2, x3, x4)
if_log2_4_in_5_ggga4(x1, x2, x3, x4)  =  if_log2_4_in_5_ggga2(x3, x4)
log2_4_out_ggga4(x1, x2, x3, x4)  =  log2_4_out_ggga1(x4)
log2_2_out_ga2(x1, x2)  =  log2_2_out_ga1(x2)
LOG2_4_IN_GGGA4(x1, x2, x3, x4)  =  LOG2_4_IN_GGGA3(x1, x2, x3)
IF_LOG2_4_IN_2_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_2_GGGA3(x2, x3, x5)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_out_g1(X)) -> LOG2_4_IN_GGGA4(Half, s_11(0_0), s_11(Acc), Y)
LOG2_4_IN_GGGA4(X, s_11(s_11(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGA5(X, Half, Acc, Y, small_1_in_g1(X))
LOG2_4_IN_GGGA4(s_11(s_11(X)), Half, Acc, Y) -> LOG2_4_IN_GGGA4(X, s_11(Half), Acc, Y)

The TRS R consists of the following rules:

small_1_in_g1(0_0) -> small_1_out_g1(0_0)
small_1_in_g1(s_11(0_0)) -> small_1_out_g1(s_11(0_0))

The argument filtering Pi contains the following mapping:
0_0  =  0_0
s_11(x1)  =  s_11(x1)
small_1_in_g1(x1)  =  small_1_in_g1(x1)
small_1_out_g1(x1)  =  small_1_out_g
LOG2_4_IN_GGGA4(x1, x2, x3, x4)  =  LOG2_4_IN_GGGA3(x1, x2, x3)
IF_LOG2_4_IN_2_GGGA5(x1, x2, x3, x4, x5)  =  IF_LOG2_4_IN_2_GGGA3(x2, x3, x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ RuleRemovalProof

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

IF_LOG2_4_IN_2_GGGA3(Half, Acc, small_1_out_g) -> LOG2_4_IN_GGGA3(Half, s_11(0_0), s_11(Acc))
LOG2_4_IN_GGGA3(X, s_11(s_11(Half)), Acc) -> IF_LOG2_4_IN_2_GGGA3(Half, Acc, small_1_in_g1(X))
LOG2_4_IN_GGGA3(s_11(s_11(X)), Half, Acc) -> LOG2_4_IN_GGGA3(X, s_11(Half), Acc)

The TRS R consists of the following rules:

small_1_in_g1(0_0) -> small_1_out_g
small_1_in_g1(s_11(0_0)) -> small_1_out_g

The set Q consists of the following terms:

small_1_in_g1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LOG2_4_IN_GGGA3, IF_LOG2_4_IN_2_GGGA3}.
By using a polynomial ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

small_1_in_g1(0_0) -> small_1_out_g
small_1_in_g1(s_11(0_0)) -> small_1_out_g

Used ordering: POLO with Polynomial interpretation:

POL(small_1_in_g1(x1)) = 2 + x1   
POL(0_0) = 0   
POL(small_1_out_g) = 1   
POL(IF_LOG2_4_IN_2_GGGA3(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(s_11(x1)) = 1 + x1   
POL(LOG2_4_IN_GGGA3(x1, x2, x3)) = x1 + 2·x2 + x3   



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ RuleRemovalProof
                        ↳ QDP
                          ↳ DependencyGraphProof

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

IF_LOG2_4_IN_2_GGGA3(Half, Acc, small_1_out_g) -> LOG2_4_IN_GGGA3(Half, s_11(0_0), s_11(Acc))
LOG2_4_IN_GGGA3(X, s_11(s_11(Half)), Acc) -> IF_LOG2_4_IN_2_GGGA3(Half, Acc, small_1_in_g1(X))
LOG2_4_IN_GGGA3(s_11(s_11(X)), Half, Acc) -> LOG2_4_IN_GGGA3(X, s_11(Half), Acc)

R is empty.
The set Q consists of the following terms:

small_1_in_g1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LOG2_4_IN_GGGA3, IF_LOG2_4_IN_2_GGGA3}.
The approximation of the Dependency Graph contains 1 SCC with 2 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ RuleRemovalProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ QDPSizeChangeProof

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

LOG2_4_IN_GGGA3(s_11(s_11(X)), Half, Acc) -> LOG2_4_IN_GGGA3(X, s_11(Half), Acc)

R is empty.
The set Q consists of the following terms:

small_1_in_g1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LOG2_4_IN_GGGA3}.
By using the subterm criterion together with the size-change analysis 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: