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



↳ Prolog
  ↳ PrologToPiTRSProof

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).

We use the technique of [30]. 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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)


Using Dependency Pairs [1,30] 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)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x4)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)
U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

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)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x4)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x4)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)
U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x4)
EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 12 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

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)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

The TRS R consists of the following rules:

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

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)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4)  =  IF_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], 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:

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

Used ordering: POLO with Polynomial interpretation [25]:

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)) = 2·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   



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof
                          ↳ QDP
                            ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 4 less nodes.