YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/EgPL3.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

log2(g,a)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) PiDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) PiDP
(9) PiDPToQDPProof [SOUND, 0 ms]
(10) QDP
(11) MRRProof [EQUIVALENT, 34 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


----------------------------------------

(0)
Obligation:
Clauses:

log2(X, Y) :- log2(X, 0, s(0), Y).
log2(s(s(X)), Half, Acc, Y) :- log2(X, s(Half), Acc, Y).
log2(X, s(s(Half)), Acc, Y) :- ','(small(X), log2(Half, s(0), s(Acc), Y)).
log2(X, Half, Y, Y) :- ','(small(X), small(Half)).
small(0).
small(s(0)).


Query: log2(g,a)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

log2_in_2: (b,f)

log2_in_4: (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_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y))
   log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X))
   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))
   U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X))
   U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half))
   U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y)
   U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y)
   U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y)
   U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ga(x1, x2)  =  log2_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

log2_in_ggga(x1, x2, x3, x4)  =  log2_in_ggga(x1, x2, x3)

s(x1)  =  s(x1)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)

small_in_g(x1)  =  small_in_g(x1)

0  =  0

small_out_g(x1)  =  small_out_g(x1)

U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)

U5_ggga(x1, x2, x3, x4)  =  U5_ggga(x1, x2, x3, x4)

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x3, x4)

log2_out_ggga(x1, x2, x3, x4)  =  log2_out_ggga(x1, x2, x3, x4)

log2_out_ga(x1, x2)  =  log2_out_ga(x1, x2)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y))
   log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X))
   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))
   U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X))
   U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half))
   U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y)
   U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y)
   U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y)
   U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ga(x1, x2)  =  log2_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

log2_in_ggga(x1, x2, x3, x4)  =  log2_in_ggga(x1, x2, x3)

s(x1)  =  s(x1)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)

small_in_g(x1)  =  small_in_g(x1)

0  =  0

small_out_g(x1)  =  small_out_g(x1)

U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)

U5_ggga(x1, x2, x3, x4)  =  U5_ggga(x1, x2, x3, x4)

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x3, x4)

log2_out_ggga(x1, x2, x3, x4)  =  log2_out_ggga(x1, x2, x3, x4)

log2_out_ga(x1, x2)  =  log2_out_ga(x1, x2)



----------------------------------------

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

   LOG2_IN_GA(X, Y) -> U1_GA(X, Y, log2_in_ggga(X, 0, s(0), Y))
   LOG2_IN_GA(X, Y) -> LOG2_IN_GGGA(X, 0, s(0), Y)
   LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> U2_GGGA(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y)
   LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X))
   LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> SMALL_IN_G(X)
   U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> U4_GGGA(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y)
   LOG2_IN_GGGA(X, Half, Y, Y) -> U5_GGGA(X, Half, Y, small_in_g(X))
   LOG2_IN_GGGA(X, Half, Y, Y) -> SMALL_IN_G(X)
   U5_GGGA(X, Half, Y, small_out_g(X)) -> U6_GGGA(X, Half, Y, small_in_g(Half))
   U5_GGGA(X, Half, Y, small_out_g(X)) -> SMALL_IN_G(Half)

The TRS R consists of the following rules:

   log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y))
   log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X))
   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))
   U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X))
   U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half))
   U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y)
   U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y)
   U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y)
   U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ga(x1, x2)  =  log2_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

log2_in_ggga(x1, x2, x3, x4)  =  log2_in_ggga(x1, x2, x3)

s(x1)  =  s(x1)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)

small_in_g(x1)  =  small_in_g(x1)

0  =  0

small_out_g(x1)  =  small_out_g(x1)

U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)

U5_ggga(x1, x2, x3, x4)  =  U5_ggga(x1, x2, x3, x4)

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x3, x4)

log2_out_ggga(x1, x2, x3, x4)  =  log2_out_ggga(x1, x2, x3, x4)

log2_out_ga(x1, x2)  =  log2_out_ga(x1, x2)

LOG2_IN_GA(x1, x2)  =  LOG2_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

LOG2_IN_GGGA(x1, x2, x3, x4)  =  LOG2_IN_GGGA(x1, x2, x3)

U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)

U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)

SMALL_IN_G(x1)  =  SMALL_IN_G(x1)

U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)

U5_GGGA(x1, x2, x3, x4)  =  U5_GGGA(x1, x2, x3, x4)

U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x3, x4)


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

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LOG2_IN_GA(X, Y) -> U1_GA(X, Y, log2_in_ggga(X, 0, s(0), Y))
   LOG2_IN_GA(X, Y) -> LOG2_IN_GGGA(X, 0, s(0), Y)
   LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> U2_GGGA(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y)
   LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X))
   LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> SMALL_IN_G(X)
   U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> U4_GGGA(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y)
   LOG2_IN_GGGA(X, Half, Y, Y) -> U5_GGGA(X, Half, Y, small_in_g(X))
   LOG2_IN_GGGA(X, Half, Y, Y) -> SMALL_IN_G(X)
   U5_GGGA(X, Half, Y, small_out_g(X)) -> U6_GGGA(X, Half, Y, small_in_g(Half))
   U5_GGGA(X, Half, Y, small_out_g(X)) -> SMALL_IN_G(Half)

The TRS R consists of the following rules:

   log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y))
   log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X))
   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))
   U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X))
   U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half))
   U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y)
   U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y)
   U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y)
   U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ga(x1, x2)  =  log2_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

log2_in_ggga(x1, x2, x3, x4)  =  log2_in_ggga(x1, x2, x3)

s(x1)  =  s(x1)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)

small_in_g(x1)  =  small_in_g(x1)

0  =  0

small_out_g(x1)  =  small_out_g(x1)

U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)

U5_ggga(x1, x2, x3, x4)  =  U5_ggga(x1, x2, x3, x4)

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x3, x4)

log2_out_ggga(x1, x2, x3, x4)  =  log2_out_ggga(x1, x2, x3, x4)

log2_out_ga(x1, x2)  =  log2_out_ga(x1, x2)

LOG2_IN_GA(x1, x2)  =  LOG2_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

LOG2_IN_GGGA(x1, x2, x3, x4)  =  LOG2_IN_GGGA(x1, x2, x3)

U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)

U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)

SMALL_IN_G(x1)  =  SMALL_IN_G(x1)

U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)

U5_GGGA(x1, x2, x3, x4)  =  U5_GGGA(x1, x2, x3, x4)

U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x3, x4)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 9 less nodes.
----------------------------------------

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X))
   U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y)
   LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y)

The TRS R consists of the following rules:

   log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y))
   log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y))
   log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X))
   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))
   U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y))
   log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X))
   U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half))
   U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y)
   U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y)
   U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y)
   U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ga(x1, x2)  =  log2_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

log2_in_ggga(x1, x2, x3, x4)  =  log2_in_ggga(x1, x2, x3)

s(x1)  =  s(x1)

U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)

U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)

small_in_g(x1)  =  small_in_g(x1)

0  =  0

small_out_g(x1)  =  small_out_g(x1)

U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)

U5_ggga(x1, x2, x3, x4)  =  U5_ggga(x1, x2, x3, x4)

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x3, x4)

log2_out_ggga(x1, x2, x3, x4)  =  log2_out_ggga(x1, x2, x3, x4)

log2_out_ga(x1, x2)  =  log2_out_ga(x1, x2)

LOG2_IN_GGGA(x1, x2, x3, x4)  =  LOG2_IN_GGGA(x1, x2, x3)

U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)


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

(7) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X))
   U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y)
   LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y)

The TRS R consists of the following rules:

   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

small_in_g(x1)  =  small_in_g(x1)

0  =  0

small_out_g(x1)  =  small_out_g(x1)

LOG2_IN_GGGA(x1, x2, x3, x4)  =  LOG2_IN_GGGA(x1, x2, x3)

U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)


We have to consider all (P,R,Pi)-chains
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(9) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LOG2_IN_GGGA(X, s(s(Half)), Acc) -> U3_GGGA(X, Half, Acc, small_in_g(X))
   U3_GGGA(X, Half, Acc, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc))
   LOG2_IN_GGGA(s(s(X)), Half, Acc) -> LOG2_IN_GGGA(X, s(Half), Acc)

The TRS R consists of the following rules:

   small_in_g(0) -> small_out_g(0)
   small_in_g(s(0)) -> small_out_g(s(0))

The set Q consists of the following terms:

   small_in_g(x0)

We have to consider all (P,Q,R)-chains.
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(11) MRRProof (EQUIVALENT)
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:

   LOG2_IN_GGGA(s(s(X)), Half, Acc) -> LOG2_IN_GGGA(X, s(Half), Acc)

Strictly oriented rules of the TRS R:

   small_in_g(0) -> small_out_g(0)

Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(LOG2_IN_GGGA(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
   POL(U3_GGGA(x_1, x_2, x_3, x_4)) = 2 + x_1 + 2*x_2 + x_3 + x_4
   POL(s(x_1)) = 1 + x_1
   POL(small_in_g(x_1)) = 2 + x_1
   POL(small_out_g(x_1)) = 1 + 2*x_1


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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LOG2_IN_GGGA(X, s(s(Half)), Acc) -> U3_GGGA(X, Half, Acc, small_in_g(X))
   U3_GGGA(X, Half, Acc, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc))

The TRS R consists of the following rules:

   small_in_g(s(0)) -> small_out_g(s(0))

The set Q consists of the following terms:

   small_in_g(x0)

We have to consider all (P,Q,R)-chains.
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(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
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(14)
TRUE
