Left Termination proof of /tmp/tmpYP7-nP/log2b-oi.pl

Left Termination of the query pattern log2(f,b) w.r.t. the given Prolog program could not be shown:

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ 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: (f,b)
log2₄: (f,b,b,b) (b,b,b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

LOG2_2_IN_AG₂(X, Y) -> IF_LOG2_2_IN_1_AG₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
LOG2_2_IN_AG₂(X, Y) -> LOG2_4_IN_AGGG₄(X, 0_0, s_1₁(0_0), Y)
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_AGGG₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_3_AGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_3_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_5_GGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> SMALL_1_IN_G₁(Half)
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_5_AGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> SMALL_1_IN_G₁(Half)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ag₂(x1, x2) = log2_2_in_ag₁(x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
if_log2_2_in_1_ag₃(x1, x2, x3) = if_log2_2_in_1_ag₁(x3)
log2_4_in_aggg₄(x1, x2, x3, x4) = log2_4_in_aggg₃(x2, x3, x4)
if_log2_4_in_1_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_1_aggg₁(x5)
if_log2_4_in_2_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_2_aggg₄(x2, x3, x4, x5)
small_1_in_a₁(x1) = small_1_in_a
small_1_out_a₁(x1) = small_1_out_a₁(x1)
if_log2_4_in_3_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_3_aggg₂(x1, x5)
log2_4_in_gggg₄(x1, x2, x3, x4) = log2_4_in_gggg₄(x1, x2, x3, x4)
if_log2_4_in_1_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_1_gggg₁(x5)
if_log2_4_in_2_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_2_gggg₄(x2, x3, x4, 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_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_3_gggg₁(x5)
if_log2_4_in_4_gggg₄(x1, x2, x3, x4) = if_log2_4_in_4_gggg₂(x2, x4)
if_log2_4_in_5_gggg₄(x1, x2, x3, x4) = if_log2_4_in_5_gggg₁(x4)
log2_4_out_gggg₄(x1, x2, x3, x4) = log2_4_out_gggg
log2_4_out_aggg₄(x1, x2, x3, x4) = log2_4_out_aggg₁(x1)
if_log2_4_in_4_aggg₄(x1, x2, x3, x4) = if_log2_4_in_4_aggg₂(x2, x4)
if_log2_4_in_5_aggg₄(x1, x2, x3, x4) = if_log2_4_in_5_aggg₂(x1, x4)
log2_2_out_ag₂(x1, x2) = log2_2_out_ag₁(x1)
SMALL_1_IN_G₁(x1) = SMALL_1_IN_G₁(x1)
IF_LOG2_4_IN_2_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_GGGG₄(x2, x3, x4, x5)
IF_LOG2_4_IN_1_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_1_AGGG₁(x5)
LOG2_4_IN_GGGG₄(x1, x2, x3, x4) = LOG2_4_IN_GGGG₄(x1, x2, x3, x4)
LOG2_2_IN_AG₂(x1, x2) = LOG2_2_IN_AG₁(x2)
IF_LOG2_4_IN_1_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_1_GGGG₁(x5)
SMALL_1_IN_A₁(x1) = SMALL_1_IN_A
LOG2_4_IN_AGGG₄(x1, x2, x3, x4) = LOG2_4_IN_AGGG₃(x2, x3, x4)
IF_LOG2_4_IN_5_AGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_5_AGGG₂(x1, x4)
IF_LOG2_4_IN_4_AGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_4_AGGG₂(x2, x4)
IF_LOG2_2_IN_1_AG₃(x1, x2, x3) = IF_LOG2_2_IN_1_AG₁(x3)
IF_LOG2_4_IN_2_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_AGGG₄(x2, x3, x4, x5)
IF_LOG2_4_IN_3_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_3_GGGG₁(x5)
IF_LOG2_4_IN_4_GGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_4_GGGG₂(x2, x4)
IF_LOG2_4_IN_3_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_3_AGGG₂(x1, x5)
IF_LOG2_4_IN_5_GGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_5_GGGG₁(x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

LOG2_2_IN_AG₂(X, Y) -> IF_LOG2_2_IN_1_AG₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
LOG2_2_IN_AG₂(X, Y) -> LOG2_4_IN_AGGG₄(X, 0_0, s_1₁(0_0), Y)
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_AGGG₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_3_AGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_3_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_5_GGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> SMALL_1_IN_G₁(Half)
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_5_AGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> SMALL_1_IN_G₁(Half)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(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
IF_LOG2_4_IN_2_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_GGGG₄(x2, x3, x4, x5)
LOG2_4_IN_GGGG₄(x1, x2, x3, x4) = LOG2_4_IN_GGGG₄(x1, x2, x3, x4)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₄(Half, Acc, Y, small_1_in_g₁(X))
IF_LOG2_4_IN_2_GGGG₄(Half, Acc, Y, small_1_out_g) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)

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 {IF_LOG2_4_IN_2_GGGG₄, LOG2_4_IN_GGGG₄}.
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(IF_LOG2_4_IN_2_GGGG₄(x₁, x₂, x₃, x₄)) = 2 + x₁ + x₂ + x₃ + x₄
POL(LOG2_4_IN_GGGG₄(x₁, x₂, x₃, x₄)) = x₁ + 2·x₂ + x₃ + x₄
POL(small_1_out_g) = 1
POL(s_1₁(x₁)) = 1 + x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₄(Half, Acc, Y, small_1_in_g₁(X))
IF_LOG2_4_IN_2_GGGG₄(Half, Acc, Y, small_1_out_g) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)

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 {IF_LOG2_4_IN_2_GGGG₄, LOG2_4_IN_GGGG₄}.
The approximation of the Dependency Graph contains 1 SCC with 2 less nodes.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)

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_GGGG₄}.
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:

LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)
The graph contains the following edges 1 > 1, 3 >= 3, 4 >= 4

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
LOG2_4_IN_AGGG₄(x1, x2, x3, x4) = LOG2_4_IN_AGGG₃(x2, x3, x4)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

  ↳ PrologToPiTRSProof

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

LOG2_4_IN_AGGG₃(Half, Acc, Y) -> LOG2_4_IN_AGGG₃(s_1₁(Half), Acc, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LOG2_4_IN_AGGG₃}.
With regard to the inferred argument filtering the predicates were used in the following modes:
log2₂: (f,b)
log2₄: (f,b,b,b) (b,b,b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

LOG2_2_IN_AG₂(X, Y) -> IF_LOG2_2_IN_1_AG₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
LOG2_2_IN_AG₂(X, Y) -> LOG2_4_IN_AGGG₄(X, 0_0, s_1₁(0_0), Y)
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_AGGG₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_3_AGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_3_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_5_GGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> SMALL_1_IN_G₁(Half)
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_5_AGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> SMALL_1_IN_G₁(Half)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ag₂(x1, x2) = log2_2_in_ag₁(x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
if_log2_2_in_1_ag₃(x1, x2, x3) = if_log2_2_in_1_ag₂(x2, x3)
log2_4_in_aggg₄(x1, x2, x3, x4) = log2_4_in_aggg₃(x2, x3, x4)
if_log2_4_in_1_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_1_aggg₄(x2, x3, x4, x5)
if_log2_4_in_2_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_2_aggg₄(x2, x3, x4, x5)
small_1_in_a₁(x1) = small_1_in_a
small_1_out_a₁(x1) = small_1_out_a₁(x1)
if_log2_4_in_3_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_3_aggg₅(x1, x2, x3, x4, x5)
log2_4_in_gggg₄(x1, x2, x3, x4) = log2_4_in_gggg₄(x1, x2, x3, x4)
if_log2_4_in_1_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_1_gggg₅(x1, x2, x3, x4, x5)
if_log2_4_in_2_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_2_gggg₅(x1, x2, x3, x4, x5)
small_1_in_g₁(x1) = small_1_in_g₁(x1)
small_1_out_g₁(x1) = small_1_out_g₁(x1)
if_log2_4_in_3_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_3_gggg₅(x1, x2, x3, x4, x5)
if_log2_4_in_4_gggg₄(x1, x2, x3, x4) = if_log2_4_in_4_gggg₄(x1, x2, x3, x4)
if_log2_4_in_5_gggg₄(x1, x2, x3, x4) = if_log2_4_in_5_gggg₄(x1, x2, x3, x4)
log2_4_out_gggg₄(x1, x2, x3, x4) = log2_4_out_gggg₄(x1, x2, x3, x4)
log2_4_out_aggg₄(x1, x2, x3, x4) = log2_4_out_aggg₄(x1, x2, x3, x4)
if_log2_4_in_4_aggg₄(x1, x2, x3, x4) = if_log2_4_in_4_aggg₃(x2, x3, x4)
if_log2_4_in_5_aggg₄(x1, x2, x3, x4) = if_log2_4_in_5_aggg₄(x1, x2, x3, x4)
log2_2_out_ag₂(x1, x2) = log2_2_out_ag₂(x1, x2)
SMALL_1_IN_G₁(x1) = SMALL_1_IN_G₁(x1)
IF_LOG2_4_IN_2_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_GGGG₅(x1, x2, x3, x4, x5)
IF_LOG2_4_IN_1_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_1_AGGG₄(x2, x3, x4, x5)
LOG2_4_IN_GGGG₄(x1, x2, x3, x4) = LOG2_4_IN_GGGG₄(x1, x2, x3, x4)
LOG2_2_IN_AG₂(x1, x2) = LOG2_2_IN_AG₁(x2)
IF_LOG2_4_IN_1_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_1_GGGG₅(x1, x2, x3, x4, x5)
SMALL_1_IN_A₁(x1) = SMALL_1_IN_A
LOG2_4_IN_AGGG₄(x1, x2, x3, x4) = LOG2_4_IN_AGGG₃(x2, x3, x4)
IF_LOG2_4_IN_5_AGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_5_AGGG₄(x1, x2, x3, x4)
IF_LOG2_4_IN_4_AGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_4_AGGG₃(x2, x3, x4)
IF_LOG2_2_IN_1_AG₃(x1, x2, x3) = IF_LOG2_2_IN_1_AG₂(x2, x3)
IF_LOG2_4_IN_2_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_AGGG₄(x2, x3, x4, x5)
IF_LOG2_4_IN_3_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_3_GGGG₅(x1, x2, x3, x4, x5)
IF_LOG2_4_IN_4_GGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_4_GGGG₄(x1, x2, x3, x4)
IF_LOG2_4_IN_3_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_3_AGGG₅(x1, x2, x3, x4, x5)
IF_LOG2_4_IN_5_GGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_5_GGGG₄(x1, x2, x3, x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

LOG2_2_IN_AG₂(X, Y) -> IF_LOG2_2_IN_1_AG₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
LOG2_2_IN_AG₂(X, Y) -> LOG2_4_IN_AGGG₄(X, 0_0, s_1₁(0_0), Y)
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_AGGG₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_3_AGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_AGGG₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> IF_LOG2_4_IN_1_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_3_GGGG₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_in_g₁(X))
LOG2_4_IN_GGGG₄(X, Half, Y, Y) -> SMALL_1_IN_G₁(X)
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> IF_LOG2_4_IN_5_GGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_GGGG₄(X, Half, Y, small_1_out_g₁(X)) -> SMALL_1_IN_G₁(Half)
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_in_a₁(X))
LOG2_4_IN_AGGG₄(X, Half, Y, Y) -> SMALL_1_IN_A₁(X)
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> IF_LOG2_4_IN_5_AGGG₄(X, Half, Y, small_1_in_g₁(Half))
IF_LOG2_4_IN_4_AGGG₄(X, Half, Y, small_1_out_a₁(X)) -> SMALL_1_IN_G₁(Half)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

The argument filtering Pi contains the following mapping:
log2_2_in_ag₂(x1, x2) = log2_2_in_ag₁(x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
if_log2_2_in_1_ag₃(x1, x2, x3) = if_log2_2_in_1_ag₂(x2, x3)
log2_4_in_aggg₄(x1, x2, x3, x4) = log2_4_in_aggg₃(x2, x3, x4)
if_log2_4_in_1_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_1_aggg₄(x2, x3, x4, x5)
if_log2_4_in_2_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_2_aggg₄(x2, x3, x4, x5)
small_1_in_a₁(x1) = small_1_in_a
small_1_out_a₁(x1) = small_1_out_a₁(x1)
if_log2_4_in_3_aggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_3_aggg₅(x1, x2, x3, x4, x5)
log2_4_in_gggg₄(x1, x2, x3, x4) = log2_4_in_gggg₄(x1, x2, x3, x4)
if_log2_4_in_1_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_1_gggg₅(x1, x2, x3, x4, x5)
if_log2_4_in_2_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_2_gggg₅(x1, x2, x3, x4, x5)
small_1_in_g₁(x1) = small_1_in_g₁(x1)
small_1_out_g₁(x1) = small_1_out_g₁(x1)
if_log2_4_in_3_gggg₅(x1, x2, x3, x4, x5) = if_log2_4_in_3_gggg₅(x1, x2, x3, x4, x5)
if_log2_4_in_4_gggg₄(x1, x2, x3, x4) = if_log2_4_in_4_gggg₄(x1, x2, x3, x4)
if_log2_4_in_5_gggg₄(x1, x2, x3, x4) = if_log2_4_in_5_gggg₄(x1, x2, x3, x4)
log2_4_out_gggg₄(x1, x2, x3, x4) = log2_4_out_gggg₄(x1, x2, x3, x4)
log2_4_out_aggg₄(x1, x2, x3, x4) = log2_4_out_aggg₄(x1, x2, x3, x4)
if_log2_4_in_4_aggg₄(x1, x2, x3, x4) = if_log2_4_in_4_aggg₃(x2, x3, x4)
if_log2_4_in_5_aggg₄(x1, x2, x3, x4) = if_log2_4_in_5_aggg₄(x1, x2, x3, x4)
log2_2_out_ag₂(x1, x2) = log2_2_out_ag₂(x1, x2)
SMALL_1_IN_G₁(x1) = SMALL_1_IN_G₁(x1)
IF_LOG2_4_IN_2_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_GGGG₅(x1, x2, x3, x4, x5)
IF_LOG2_4_IN_1_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_1_AGGG₄(x2, x3, x4, x5)
LOG2_4_IN_GGGG₄(x1, x2, x3, x4) = LOG2_4_IN_GGGG₄(x1, x2, x3, x4)
LOG2_2_IN_AG₂(x1, x2) = LOG2_2_IN_AG₁(x2)
IF_LOG2_4_IN_1_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_1_GGGG₅(x1, x2, x3, x4, x5)
SMALL_1_IN_A₁(x1) = SMALL_1_IN_A
LOG2_4_IN_AGGG₄(x1, x2, x3, x4) = LOG2_4_IN_AGGG₃(x2, x3, x4)
IF_LOG2_4_IN_5_AGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_5_AGGG₄(x1, x2, x3, x4)
IF_LOG2_4_IN_4_AGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_4_AGGG₃(x2, x3, x4)
IF_LOG2_2_IN_1_AG₃(x1, x2, x3) = IF_LOG2_2_IN_1_AG₂(x2, x3)
IF_LOG2_4_IN_2_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_2_AGGG₄(x2, x3, x4, x5)
IF_LOG2_4_IN_3_GGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_3_GGGG₅(x1, x2, x3, x4, x5)
IF_LOG2_4_IN_4_GGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_4_GGGG₄(x1, x2, x3, x4)
IF_LOG2_4_IN_3_AGGG₅(x1, x2, x3, x4, x5) = IF_LOG2_4_IN_3_AGGG₅(x1, x2, x3, x4, x5)
IF_LOG2_4_IN_5_GGGG₄(x1, x2, x3, x4) = IF_LOG2_4_IN_5_GGGG₄(x1, x2, x3, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 18 less nodes.

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(X, s_1₁(Half), Acc, Y)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

              ↳ PiDP

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

LOG2_4_IN_GGGG₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_in_g₁(X))
IF_LOG2_4_IN_2_GGGG₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> LOG2_4_IN_GGGG₄(Half, s_1₁(0_0), s_1₁(Acc), Y)
LOG2_4_IN_GGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_GGGG₄(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))

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

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

LOG2_4_IN_AGGG₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> LOG2_4_IN_AGGG₄(X, s_1₁(Half), Acc, Y)

The TRS R consists of the following rules:

log2_2_in_ag₂(X, Y) -> if_log2_2_in_1_ag₃(X, Y, log2_4_in_aggg₄(X, 0_0, s_1₁(0_0), Y))
log2_4_in_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_in_aggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_in_a₁(X))
small_1_in_a₁(0_0) -> small_1_out_a₁(0_0)
small_1_in_a₁(s_1₁(0_0)) -> small_1_out_a₁(s_1₁(0_0))
if_log2_4_in_2_aggg₅(X, Half, Acc, Y, small_1_out_a₁(X)) -> if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y) -> if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(X, s_1₁(Half), Acc, Y))
log2_4_in_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y) -> if_log2_4_in_2_gggg₅(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_gggg₅(X, Half, Acc, Y, small_1_out_g₁(X)) -> if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_in_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y))
log2_4_in_gggg₄(X, Half, Y, Y) -> if_log2_4_in_4_gggg₄(X, Half, Y, small_1_in_g₁(X))
if_log2_4_in_4_gggg₄(X, Half, Y, small_1_out_g₁(X)) -> if_log2_4_in_5_gggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_gggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_gggg₄(X, Half, Y, Y)
if_log2_4_in_3_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_gggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
if_log2_4_in_1_gggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_gggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_4_in_3_aggg₅(X, Half, Acc, Y, log2_4_out_gggg₄(Half, s_1₁(0_0), s_1₁(Acc), Y)) -> log2_4_out_aggg₄(X, s_1₁(s_1₁(Half)), Acc, Y)
log2_4_in_aggg₄(X, Half, Y, Y) -> if_log2_4_in_4_aggg₄(X, Half, Y, small_1_in_a₁(X))
if_log2_4_in_4_aggg₄(X, Half, Y, small_1_out_a₁(X)) -> if_log2_4_in_5_aggg₄(X, Half, Y, small_1_in_g₁(Half))
if_log2_4_in_5_aggg₄(X, Half, Y, small_1_out_g₁(Half)) -> log2_4_out_aggg₄(X, Half, Y, Y)
if_log2_4_in_1_aggg₅(X, Half, Acc, Y, log2_4_out_aggg₄(X, s_1₁(Half), Acc, Y)) -> log2_4_out_aggg₄(s_1₁(s_1₁(X)), Half, Acc, Y)
if_log2_2_in_1_ag₃(X, Y, log2_4_out_aggg₄(X, 0_0, s_1₁(0_0), Y)) -> log2_2_out_ag₂(X, Y)