Left Termination of the query pattern times_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

even(0, true).
even(s(0), false).
even(s(s(X)), B) :- even(X, B).
half(0, 0).
half(s(s(X)), s(Y)) :- half(X, Y).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
times(0, Y, 0).
times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z)).
if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z))).
if(false, s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)).

Queries:

times(g,g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
times_in: (b,b,f)
even_in: (b,f)
if_in: (b,b,b,f)
half_in: (b,f)
plus_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
TIMES_IN_GGA(s(X), Y, Z) → EVEN_IN_GA(s(X), B)
EVEN_IN_GA(s(s(X)), B) → U1_GA(X, B, even_in_ga(X, B))
EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
IF_IN_GGGA(true, s(X), Y, Z) → HALF_IN_GA(s(X), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U2_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z))
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → PLUS_IN_GGA(Y1, Y1, Z)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)
IF_IN_GGGA(false, s(X), Y, Z) → U9_GGGA(X, Y, Z, times_in_gga(X, Y, U))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z))
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → PLUS_IN_GGA(Y, U, Z)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x4)
U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x4)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x2, x4)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x4)

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

(4) Obligation:

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

TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
TIMES_IN_GGA(s(X), Y, Z) → EVEN_IN_GA(s(X), B)
EVEN_IN_GA(s(s(X)), B) → U1_GA(X, B, even_in_ga(X, B))
EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
IF_IN_GGGA(true, s(X), Y, Z) → HALF_IN_GA(s(X), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U2_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z))
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → PLUS_IN_GGA(Y1, Y1, Z)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)
IF_IN_GGGA(false, s(X), Y, Z) → U9_GGGA(X, Y, Z, times_in_gga(X, Y, U))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z))
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → PLUS_IN_GGA(Y, U, Z)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x4)
U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x4)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x2, x4)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_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:

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_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:

PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)

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:

• PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) TRUE

(14) Obligation:

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)

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:

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)

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:

HALF_IN_GA(s(s(X))) → HALF_IN_GA(X)

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:

• HALF_IN_GA(s(s(X))) → HALF_IN_GA(X)
  The graph contains the following edges 1 > 1

(20) TRUE

(21) Obligation:

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

EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)
EVEN_IN_GA(x1, x2)  =  EVEN_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:

EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
EVEN_IN_GA(x1, x2)  =  EVEN_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:

EVEN_IN_GA(s(s(X))) → EVEN_IN_GA(X)

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

(26) 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:

• EVEN_IN_GA(s(s(X))) → EVEN_IN_GA(X)
  The graph contains the following edges 1 > 1

(27) TRUE

(28) Obligation:

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

U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)

The TRS R consists of the following rules:

half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
half_in_ga(0, 0) → half_out_ga(0, 0)
even_in_ga(0, true) → even_out_ga(0, true)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
true  =  true
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
false  =  false
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)

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:

U4_GGA(X, Y, even_out_ga(B)) → IF_IN_GGGA(B, s(X), Y)
IF_IN_GGGA(true, s(X), Y) → U6_GGGA(Y, half_in_ga(s(X)))
U6_GGGA(Y, half_out_ga(X1)) → TIMES_IN_GGA(X1, Y)
TIMES_IN_GGA(s(X), Y) → U4_GGA(X, Y, even_in_ga(s(X)))
IF_IN_GGGA(false, s(X), Y) → TIMES_IN_GGA(X, Y)

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U2_ga(half_in_ga(X))
even_in_ga(s(0)) → even_out_ga(false)
even_in_ga(s(s(X))) → U1_ga(even_in_ga(X))
U2_ga(half_out_ga(Y)) → half_out_ga(s(Y))
U1_ga(even_out_ga(B)) → even_out_ga(B)
half_in_ga(0) → half_out_ga(0)
even_in_ga(0) → even_out_ga(true)

The set Q consists of the following terms:

half_in_ga(x0)
even_in_ga(x0)
U2_ga(x0)
U1_ga(x0)

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

(33) 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:

IF_IN_GGGA(false, s(X), Y) → TIMES_IN_GGA(X, Y)

Strictly oriented rules of the TRS R:

half_in_ga(s(s(X))) → U2_ga(half_in_ga(X))
even_in_ga(s(0)) → even_out_ga(false)
even_in_ga(s(s(X))) → U1_ga(even_in_ga(X))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IF_IN_GGGA(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(TIMES_IN_GGA(x₁, x₂)) = 2·x₁ + x₂   
POL(U1_ga(x₁)) = 2·x₁   
POL(U2_ga(x₁)) = 2 + 2·x₁   
POL(U4_GGA(x₁, x₂, x₃)) = 1 + 2·x₁ + x₂ + x₃   
POL(U6_GGGA(x₁, x₂)) = x₁ + x₂   
POL(even_in_ga(x₁)) = x₁   
POL(even_out_ga(x₁)) = x₁   
POL(false) = 0   
POL(half_in_ga(x₁)) = x₁   
POL(half_out_ga(x₁)) = 2·x₁   
POL(s(x₁)) = 1 + 2·x₁   
POL(true) = 0   

(34) Obligation:

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

U4_GGA(X, Y, even_out_ga(B)) → IF_IN_GGGA(B, s(X), Y)
IF_IN_GGGA(true, s(X), Y) → U6_GGGA(Y, half_in_ga(s(X)))
U6_GGGA(Y, half_out_ga(X1)) → TIMES_IN_GGA(X1, Y)
TIMES_IN_GGA(s(X), Y) → U4_GGA(X, Y, even_in_ga(s(X)))

The TRS R consists of the following rules:

U2_ga(half_out_ga(Y)) → half_out_ga(s(Y))
U1_ga(even_out_ga(B)) → even_out_ga(B)
half_in_ga(0) → half_out_ga(0)
even_in_ga(0) → even_out_ga(true)

The set Q consists of the following terms:

half_in_ga(x0)
even_in_ga(x0)
U2_ga(x0)
U1_ga(x0)

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 4 less nodes.

(36) TRUE

(37) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
times_in: (b,b,f)
even_in: (b,f)
if_in: (b,b,b,f)
half_in: (b,f)
plus_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(38) Obligation:

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

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)

(39) DependencyPairsProof (EQUIVALENT transformation)

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

TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
TIMES_IN_GGA(s(X), Y, Z) → EVEN_IN_GA(s(X), B)
EVEN_IN_GA(s(s(X)), B) → U1_GA(X, B, even_in_ga(X, B))
EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
IF_IN_GGGA(true, s(X), Y, Z) → HALF_IN_GA(s(X), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U2_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z))
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → PLUS_IN_GGA(Y1, Y1, Z)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)
IF_IN_GGGA(false, s(X), Y, Z) → U9_GGGA(X, Y, Z, times_in_gga(X, Y, U))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z))
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → PLUS_IN_GGA(Y, U, Z)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x1, x2, x4)
U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x1, x2, x4)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x1, x2, x4)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x1, x2, x4)

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

(40) Obligation:

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

TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
TIMES_IN_GGA(s(X), Y, Z) → EVEN_IN_GA(s(X), B)
EVEN_IN_GA(s(s(X)), B) → U1_GA(X, B, even_in_ga(X, B))
EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
IF_IN_GGGA(true, s(X), Y, Z) → HALF_IN_GA(s(X), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U2_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z))
U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) → PLUS_IN_GGA(Y1, Y1, Z)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)
IF_IN_GGGA(false, s(X), Y, Z) → U9_GGGA(X, Y, Z, times_in_gga(X, Y, U))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z))
U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) → PLUS_IN_GGA(Y, U, Z)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x1, x2, x4)
U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x1, x2, x4)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x1, x2, x4)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x1, x2, x4)

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

(41) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.

(42) Complex Obligation (AND)

(43) Obligation:

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

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)

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

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)

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

(46) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(47) Obligation:

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

PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)

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

(48) 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:

• PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 >= 2

(49) TRUE

(50) Obligation:

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)

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

(53) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(54) Obligation:

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

HALF_IN_GA(s(s(X))) → HALF_IN_GA(X)

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

(55) 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:

• HALF_IN_GA(s(s(X))) → HALF_IN_GA(X)
  The graph contains the following edges 1 > 1

(56) TRUE

(57) Obligation:

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

EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)

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

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

EVEN_IN_GA(s(s(X)), B) → EVEN_IN_GA(X, B)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)

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

(60) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(61) Obligation:

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

EVEN_IN_GA(s(s(X))) → EVEN_IN_GA(X)

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

(62) 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:

• EVEN_IN_GA(s(s(X))) → EVEN_IN_GA(X)
  The graph contains the following edges 1 > 1

(63) TRUE

(64) Obligation:

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

U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)

The TRS R consists of the following rules:

times_in_gga(0, Y, 0) → times_out_gga(0, Y, 0)
times_in_gga(s(X), Y, Z) → U4_gga(X, Y, Z, even_in_ga(s(X), B))
even_in_ga(0, true) → even_out_ga(0, true)
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
U4_gga(X, Y, Z, even_out_ga(s(X), B)) → U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
if_in_ggga(true, s(X), Y, Z) → U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) → U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) → U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
plus_in_gga(0, Y, Y) → plus_out_gga(0, Y, Y)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) → if_out_ggga(true, s(X), Y, Z)
if_in_ggga(false, s(X), Y, Z) → U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) → U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) → if_out_ggga(false, s(X), Y, Z)
U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) → times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
0  =  0
times_out_gga(x1, x2, x3)  =  times_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4)  =  if_in_ggga(x1, x2, x3)
true  =  true
U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x1, x2, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x1, x2, x4)
U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x1, x2, x4)
plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
plus_out_gga(x1, x2, x3)  =  plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
if_out_ggga(x1, x2, x3, x4)  =  if_out_ggga(x1, x2, x3, x4)
false  =  false
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x4)

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

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

U4_GGA(X, Y, Z, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y, Z)
IF_IN_GGGA(true, s(X), Y, Z) → U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y, Y1)
TIMES_IN_GGA(s(X), Y, Z) → U4_GGA(X, Y, Z, even_in_ga(s(X), B))
IF_IN_GGGA(false, s(X), Y, Z) → TIMES_IN_GGA(X, Y, U)

The TRS R consists of the following rules:

half_in_ga(s(s(X)), s(Y)) → U2_ga(X, Y, half_in_ga(X, Y))
even_in_ga(s(0), false) → even_out_ga(s(0), false)
even_in_ga(s(s(X)), B) → U1_ga(X, B, even_in_ga(X, B))
U2_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U1_ga(X, B, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
half_in_ga(0, 0) → half_out_ga(0, 0)
even_in_ga(0, true) → even_out_ga(0, true)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
even_in_ga(x1, x2)  =  even_in_ga(x1)
even_out_ga(x1, x2)  =  even_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
true  =  true
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
false  =  false
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x4)

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

(67) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(68) Obligation:

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

U4_GGA(X, Y, even_out_ga(s(X), B)) → IF_IN_GGGA(B, s(X), Y)
IF_IN_GGGA(true, s(X), Y) → U6_GGGA(X, Y, half_in_ga(s(X)))
U6_GGGA(X, Y, half_out_ga(s(X), X1)) → TIMES_IN_GGA(X1, Y)
TIMES_IN_GGA(s(X), Y) → U4_GGA(X, Y, even_in_ga(s(X)))
IF_IN_GGGA(false, s(X), Y) → TIMES_IN_GGA(X, Y)

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U2_ga(X, half_in_ga(X))
even_in_ga(s(0)) → even_out_ga(s(0), false)
even_in_ga(s(s(X))) → U1_ga(X, even_in_ga(X))
U2_ga(X, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U1_ga(X, even_out_ga(X, B)) → even_out_ga(s(s(X)), B)
half_in_ga(0) → half_out_ga(0, 0)
even_in_ga(0) → even_out_ga(0, true)

The set Q consists of the following terms:

half_in_ga(x0)
even_in_ga(x0)
U2_ga(x0, x1)
U1_ga(x0, x1)

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