Left Termination proof of /tmp/tmpvBP8Qo/times2.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

even₂(0₀, true₀).
even₂(s₁(0₀), false₀).
even₂(s₁²(X), B) :- even₂(X, B).
half₂(0₀, 0₀).
half₂(s₁²(X), s₁(Y)) :- half₂(X, Y).
plus₃(0₀, Y, Y).
plus₃(s₁(X), Y, s₁(Z)) :- plus₃(X, Y, Z).
times₃(0₀, Y, 0₀).
times₃(s₁(X), Y, Z) :- even₂(s₁(X), B), if₄(B, s₁(X), Y, Z).
if₄(true₀, s₁(X), Y, Z) :- half₂(s₁(X), X1), times₃(X1, Y, Y1), plus₃(Y1, Y1, Z).
if₄(false₀, s₁(X), Y, Z) :- times₃(X, Y, U), plus₃(Y, U, Z).

With regard to the inferred argument filtering the predicates were used in the following modes:
times₃: (b,b,f)
even₂: (b,f)
if₄: (b,b,b,f)
half₂: (b,f)
plus₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> EVEN_2_IN_GA₂(s_1₁(X), B)
EVEN_2_IN_GA₂(s_1₁(s_1₁(X)), B) -> IF_EVEN_2_IN_1_GA₃(X, B, even_2_in_ga₂(X, B))
EVEN_2_IN_GA₂(s_1₁(s_1₁(X)), B) -> EVEN_2_IN_GA₂(X, B)
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> IF_TIMES_3_IN_2_GGA₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> IF_4_IN_GGGA₄(B, s_1₁(X), Y, Z)
IF_4_IN_GGGA₄(true_0, s_1₁(X), Y, Z) -> IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
IF_4_IN_GGGA₄(true_0, s_1₁(X), Y, Z) -> HALF_2_IN_GA₂(s_1₁(X), X1)
HALF_2_IN_GA₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> IF_HALF_2_IN_1_GA₃(X, Y, half_2_in_ga₂(X, Y))
HALF_2_IN_GA₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> HALF_2_IN_GA₂(X, Y)
IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> IF_IF_4_IN_2_GGGA₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> TIMES_3_IN_GGA₃(X1, Y, Y1)
IF_IF_4_IN_2_GGGA₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> IF_IF_4_IN_3_GGGA₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
IF_IF_4_IN_2_GGGA₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> PLUS_3_IN_GGA₃(Y1, Y1, Z)
PLUS_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_PLUS_3_IN_1_GGA₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
PLUS_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> PLUS_3_IN_GGA₃(X, Y, Z)
IF_4_IN_GGGA₄(false_0, s_1₁(X), Y, Z) -> IF_IF_4_IN_4_GGGA₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
IF_4_IN_GGGA₄(false_0, s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)
IF_IF_4_IN_4_GGGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> IF_IF_4_IN_5_GGGA₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
IF_IF_4_IN_4_GGGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> PLUS_3_IN_GGA₃(Y, U, Z)

The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga₃(x1, x2, x3) = times_3_in_gga₂(x1, x2)
0_0 = 0_0
true_0 = true_0
s_1₁(x1) = s_1₁(x1)
false_0 = false_0
times_3_out_gga₃(x1, x2, x3) = times_3_out_gga₁(x3)
if_times_3_in_1_gga₄(x1, x2, x3, x4) = if_times_3_in_1_gga₃(x1, x2, x4)
even_2_in_ga₂(x1, x2) = even_2_in_ga₁(x1)
even_2_out_ga₂(x1, x2) = even_2_out_ga₁(x2)
if_even_2_in_1_ga₃(x1, x2, x3) = if_even_2_in_1_ga₁(x3)
if_times_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_times_3_in_2_gga₁(x5)
if_4_in_ggga₄(x1, x2, x3, x4) = if_4_in_ggga₃(x1, x2, x3)
if_if_4_in_1_ggga₄(x1, x2, x3, x4) = if_if_4_in_1_ggga₂(x2, x4)
half_2_in_ga₂(x1, x2) = half_2_in_ga₁(x1)
half_2_out_ga₂(x1, x2) = half_2_out_ga₁(x2)
if_half_2_in_1_ga₃(x1, x2, x3) = if_half_2_in_1_ga₁(x3)
if_if_4_in_2_ggga₅(x1, x2, x3, x4, x5) = if_if_4_in_2_ggga₁(x5)
if_if_4_in_3_ggga₅(x1, x2, x3, x4, x5) = if_if_4_in_3_ggga₁(x5)
plus_3_in_gga₃(x1, x2, x3) = plus_3_in_gga₂(x1, x2)
plus_3_out_gga₃(x1, x2, x3) = plus_3_out_gga₁(x3)
if_plus_3_in_1_gga₄(x1, x2, x3, x4) = if_plus_3_in_1_gga₁(x4)
if_4_out_ggga₄(x1, x2, x3, x4) = if_4_out_ggga₁(x4)
if_if_4_in_4_ggga₄(x1, x2, x3, x4) = if_if_4_in_4_ggga₂(x2, x4)
if_if_4_in_5_ggga₅(x1, x2, x3, x4, x5) = if_if_4_in_5_ggga₁(x5)
IF_TIMES_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_GGA₃(x1, x2, x4)
IF_4_IN_GGGA₄(x1, x2, x3, x4) = IF_4_IN_GGGA₃(x1, x2, x3)
IF_IF_4_IN_3_GGGA₅(x1, x2, x3, x4, x5) = IF_IF_4_IN_3_GGGA₁(x5)
TIMES_3_IN_GGA₃(x1, x2, x3) = TIMES_3_IN_GGA₂(x1, x2)
IF_IF_4_IN_1_GGGA₄(x1, x2, x3, x4) = IF_IF_4_IN_1_GGGA₂(x2, x4)
IF_HALF_2_IN_1_GA₃(x1, x2, x3) = IF_HALF_2_IN_1_GA₁(x3)
IF_EVEN_2_IN_1_GA₃(x1, x2, x3) = IF_EVEN_2_IN_1_GA₁(x3)
IF_IF_4_IN_2_GGGA₅(x1, x2, x3, x4, x5) = IF_IF_4_IN_2_GGGA₁(x5)
IF_TIMES_3_IN_2_GGA₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_2_GGA₁(x5)
HALF_2_IN_GA₂(x1, x2) = HALF_2_IN_GA₁(x1)
IF_IF_4_IN_5_GGGA₅(x1, x2, x3, x4, x5) = IF_IF_4_IN_5_GGGA₁(x5)
IF_PLUS_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_PLUS_3_IN_1_GGA₁(x4)
EVEN_2_IN_GA₂(x1, x2) = EVEN_2_IN_GA₁(x1)
PLUS_3_IN_GGA₃(x1, x2, x3) = PLUS_3_IN_GGA₂(x1, x2)
IF_IF_4_IN_4_GGGA₄(x1, x2, x3, x4) = IF_IF_4_IN_4_GGGA₂(x2, x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> EVEN_2_IN_GA₂(s_1₁(X), B)
EVEN_2_IN_GA₂(s_1₁(s_1₁(X)), B) -> IF_EVEN_2_IN_1_GA₃(X, B, even_2_in_ga₂(X, B))
EVEN_2_IN_GA₂(s_1₁(s_1₁(X)), B) -> EVEN_2_IN_GA₂(X, B)
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> IF_TIMES_3_IN_2_GGA₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> IF_4_IN_GGGA₄(B, s_1₁(X), Y, Z)
IF_4_IN_GGGA₄(true_0, s_1₁(X), Y, Z) -> IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
IF_4_IN_GGGA₄(true_0, s_1₁(X), Y, Z) -> HALF_2_IN_GA₂(s_1₁(X), X1)
HALF_2_IN_GA₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> IF_HALF_2_IN_1_GA₃(X, Y, half_2_in_ga₂(X, Y))
HALF_2_IN_GA₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> HALF_2_IN_GA₂(X, Y)
IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> IF_IF_4_IN_2_GGGA₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> TIMES_3_IN_GGA₃(X1, Y, Y1)
IF_IF_4_IN_2_GGGA₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> IF_IF_4_IN_3_GGGA₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
IF_IF_4_IN_2_GGGA₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> PLUS_3_IN_GGA₃(Y1, Y1, Z)
PLUS_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_PLUS_3_IN_1_GGA₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
PLUS_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> PLUS_3_IN_GGA₃(X, Y, Z)
IF_4_IN_GGGA₄(false_0, s_1₁(X), Y, Z) -> IF_IF_4_IN_4_GGGA₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
IF_4_IN_GGGA₄(false_0, s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)
IF_IF_4_IN_4_GGGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> IF_IF_4_IN_5_GGGA₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
IF_IF_4_IN_4_GGGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> PLUS_3_IN_GGA₃(Y, U, Z)

The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PLUS_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> PLUS_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PLUS_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> PLUS_3_IN_GGA₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
PLUS_3_IN_GGA₃(x1, x2, x3) = PLUS_3_IN_GGA₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PLUS_3_IN_GGA₂(s_1₁(X), Y) -> PLUS_3_IN_GGA₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {PLUS_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

PLUS_3_IN_GGA₂(s_1₁(X), Y) -> PLUS_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

HALF_2_IN_GA₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> HALF_2_IN_GA₂(X, Y)

The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

HALF_2_IN_GA₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> HALF_2_IN_GA₂(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
HALF_2_IN_GA₂(x1, x2) = HALF_2_IN_GA₁(x1)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

HALF_2_IN_GA₁(s_1₁(s_1₁(X))) -> HALF_2_IN_GA₁(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {HALF_2_IN_GA₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

HALF_2_IN_GA₁(s_1₁(s_1₁(X))) -> HALF_2_IN_GA₁(X)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

EVEN_2_IN_GA₂(s_1₁(s_1₁(X)), B) -> EVEN_2_IN_GA₂(X, B)

The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

EVEN_2_IN_GA₂(s_1₁(s_1₁(X)), B) -> EVEN_2_IN_GA₂(X, B)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
EVEN_2_IN_GA₂(x1, x2) = EVEN_2_IN_GA₁(x1)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

EVEN_2_IN_GA₁(s_1₁(s_1₁(X))) -> EVEN_2_IN_GA₁(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {EVEN_2_IN_GA₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

EVEN_2_IN_GA₁(s_1₁(s_1₁(X))) -> EVEN_2_IN_GA₁(X)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

IF_4_IN_GGGA₄(false_0, s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)
IF_4_IN_GGGA₄(true_0, s_1₁(X), Y, Z) -> IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> TIMES_3_IN_GGA₃(X1, Y, Y1)
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> IF_4_IN_GGGA₄(B, s_1₁(X), Y, Z)
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))

The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga₃(x1, x2, x3) = times_3_in_gga₂(x1, x2)
0_0 = 0_0
true_0 = true_0
s_1₁(x1) = s_1₁(x1)
false_0 = false_0
times_3_out_gga₃(x1, x2, x3) = times_3_out_gga₁(x3)
if_times_3_in_1_gga₄(x1, x2, x3, x4) = if_times_3_in_1_gga₃(x1, x2, x4)
even_2_in_ga₂(x1, x2) = even_2_in_ga₁(x1)
even_2_out_ga₂(x1, x2) = even_2_out_ga₁(x2)
if_even_2_in_1_ga₃(x1, x2, x3) = if_even_2_in_1_ga₁(x3)
if_times_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_times_3_in_2_gga₁(x5)
if_4_in_ggga₄(x1, x2, x3, x4) = if_4_in_ggga₃(x1, x2, x3)
if_if_4_in_1_ggga₄(x1, x2, x3, x4) = if_if_4_in_1_ggga₂(x2, x4)
half_2_in_ga₂(x1, x2) = half_2_in_ga₁(x1)
half_2_out_ga₂(x1, x2) = half_2_out_ga₁(x2)
if_half_2_in_1_ga₃(x1, x2, x3) = if_half_2_in_1_ga₁(x3)
if_if_4_in_2_ggga₅(x1, x2, x3, x4, x5) = if_if_4_in_2_ggga₁(x5)
if_if_4_in_3_ggga₅(x1, x2, x3, x4, x5) = if_if_4_in_3_ggga₁(x5)
plus_3_in_gga₃(x1, x2, x3) = plus_3_in_gga₂(x1, x2)
plus_3_out_gga₃(x1, x2, x3) = plus_3_out_gga₁(x3)
if_plus_3_in_1_gga₄(x1, x2, x3, x4) = if_plus_3_in_1_gga₁(x4)
if_4_out_ggga₄(x1, x2, x3, x4) = if_4_out_ggga₁(x4)
if_if_4_in_4_ggga₄(x1, x2, x3, x4) = if_if_4_in_4_ggga₂(x2, x4)
if_if_4_in_5_ggga₅(x1, x2, x3, x4, x5) = if_if_4_in_5_ggga₁(x5)
IF_TIMES_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_GGA₃(x1, x2, x4)
IF_4_IN_GGGA₄(x1, x2, x3, x4) = IF_4_IN_GGGA₃(x1, x2, x3)
TIMES_3_IN_GGA₃(x1, x2, x3) = TIMES_3_IN_GGA₂(x1, x2)
IF_IF_4_IN_1_GGGA₄(x1, x2, x3, x4) = IF_IF_4_IN_1_GGGA₂(x2, x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

IF_4_IN_GGGA₄(false_0, s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)
IF_4_IN_GGGA₄(true_0, s_1₁(X), Y, Z) -> IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
IF_IF_4_IN_1_GGGA₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> TIMES_3_IN_GGA₃(X1, Y, Y1)
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> IF_4_IN_GGGA₄(B, s_1₁(X), Y, Z)
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))

The TRS R consists of the following rules:

half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)

The argument filtering Pi contains the following mapping:
0_0 = 0_0
true_0 = true_0
s_1₁(x1) = s_1₁(x1)
false_0 = false_0
even_2_in_ga₂(x1, x2) = even_2_in_ga₁(x1)
even_2_out_ga₂(x1, x2) = even_2_out_ga₁(x2)
if_even_2_in_1_ga₃(x1, x2, x3) = if_even_2_in_1_ga₁(x3)
half_2_in_ga₂(x1, x2) = half_2_in_ga₁(x1)
half_2_out_ga₂(x1, x2) = half_2_out_ga₁(x2)
if_half_2_in_1_ga₃(x1, x2, x3) = if_half_2_in_1_ga₁(x3)
IF_TIMES_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_GGA₃(x1, x2, x4)
IF_4_IN_GGGA₄(x1, x2, x3, x4) = IF_4_IN_GGGA₃(x1, x2, x3)
TIMES_3_IN_GGA₃(x1, x2, x3) = TIMES_3_IN_GGA₂(x1, x2)
IF_IF_4_IN_1_GGGA₄(x1, x2, x3, x4) = IF_IF_4_IN_1_GGGA₂(x2, x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

  ↳ PrologToPiTRSProof

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

IF_4_IN_GGGA₃(false_0, s_1₁(X), Y) -> TIMES_3_IN_GGA₂(X, Y)
IF_4_IN_GGGA₃(true_0, s_1₁(X), Y) -> IF_IF_4_IN_1_GGGA₂(Y, half_2_in_ga₁(s_1₁(X)))
IF_IF_4_IN_1_GGGA₂(Y, half_2_out_ga₁(X1)) -> TIMES_3_IN_GGA₂(X1, Y)
IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_out_ga₁(B)) -> IF_4_IN_GGGA₃(B, s_1₁(X), Y)
TIMES_3_IN_GGA₂(s_1₁(X), Y) -> IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_in_ga₁(s_1₁(X)))

The TRS R consists of the following rules:

half_2_in_ga₁(s_1₁(s_1₁(X))) -> if_half_2_in_1_ga₁(half_2_in_ga₁(X))
even_2_in_ga₁(s_1₁(0_0)) -> even_2_out_ga₁(false_0)
even_2_in_ga₁(s_1₁(s_1₁(X))) -> if_even_2_in_1_ga₁(even_2_in_ga₁(X))
if_half_2_in_1_ga₁(half_2_out_ga₁(Y)) -> half_2_out_ga₁(s_1₁(Y))
if_even_2_in_1_ga₁(even_2_out_ga₁(B)) -> even_2_out_ga₁(B)
half_2_in_ga₁(0_0) -> half_2_out_ga₁(0_0)
even_2_in_ga₁(0_0) -> even_2_out_ga₁(true_0)

The set Q consists of the following terms:

half_2_in_ga₁(x0)
even_2_in_ga₁(x0)
if_half_2_in_1_ga₁(x0)
if_even_2_in_1_ga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {TIMES_3_IN_GGA₂, IF_4_IN_GGGA₃, IF_IF_4_IN_1_GGGA₂, IF_TIMES_3_IN_1_GGA₃}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

IF_4_IN_GGGA₃(false_0, s_1₁(X), Y) -> TIMES_3_IN_GGA₂(X, Y)

The remaining Dependency Pairs were at least non-strictly be oriented.

IF_4_IN_GGGA₃(true_0, s_1₁(X), Y) -> IF_IF_4_IN_1_GGGA₂(Y, half_2_in_ga₁(s_1₁(X)))
IF_IF_4_IN_1_GGGA₂(Y, half_2_out_ga₁(X1)) -> TIMES_3_IN_GGA₂(X1, Y)
IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_out_ga₁(B)) -> IF_4_IN_GGGA₃(B, s_1₁(X), Y)
TIMES_3_IN_GGA₂(s_1₁(X), Y) -> IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_in_ga₁(s_1₁(X)))

With the implicit AFS we had to orient the following set of usable rules non-strictly.

if_half_2_in_1_ga₁(half_2_out_ga₁(Y)) -> half_2_out_ga₁(s_1₁(Y))
half_2_in_ga₁(0_0) -> half_2_out_ga₁(0_0)
half_2_in_ga₁(s_1₁(s_1₁(X))) -> if_half_2_in_1_ga₁(half_2_in_ga₁(X))

Used ordering: POLO with Polynomial interpretation:

POL(0_0) = 0
POL(if_half_2_in_1_ga₁(x₁)) = 1 + x₁
POL(TIMES_3_IN_GGA₂(x₁, x₂)) = x₁ + x₂
POL(false_0) = 0
POL(half_2_in_ga₁(x₁)) = x₁
POL(true_0) = 0
POL(IF_TIMES_3_IN_1_GGA₃(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(half_2_out_ga₁(x₁)) = x₁
POL(IF_4_IN_GGGA₃(x₁, x₂, x₃)) = x₂ + x₃
POL(if_even_2_in_1_ga₁(x₁)) = 0
POL(even_2_in_ga₁(x₁)) = 0
POL(even_2_out_ga₁(x₁)) = 0
POL(IF_IF_4_IN_1_GGGA₂(x₁, x₂)) = x₁ + x₂
POL(s_1₁(x₁)) = 1 + x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

                          ↳ QDP

                            ↳ QDPPoloProof

  ↳ PrologToPiTRSProof

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

IF_4_IN_GGGA₃(true_0, s_1₁(X), Y) -> IF_IF_4_IN_1_GGGA₂(Y, half_2_in_ga₁(s_1₁(X)))
IF_IF_4_IN_1_GGGA₂(Y, half_2_out_ga₁(X1)) -> TIMES_3_IN_GGA₂(X1, Y)
IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_out_ga₁(B)) -> IF_4_IN_GGGA₃(B, s_1₁(X), Y)
TIMES_3_IN_GGA₂(s_1₁(X), Y) -> IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_in_ga₁(s_1₁(X)))

The TRS R consists of the following rules:

half_2_in_ga₁(s_1₁(s_1₁(X))) -> if_half_2_in_1_ga₁(half_2_in_ga₁(X))
even_2_in_ga₁(s_1₁(0_0)) -> even_2_out_ga₁(false_0)
even_2_in_ga₁(s_1₁(s_1₁(X))) -> if_even_2_in_1_ga₁(even_2_in_ga₁(X))
if_half_2_in_1_ga₁(half_2_out_ga₁(Y)) -> half_2_out_ga₁(s_1₁(Y))
if_even_2_in_1_ga₁(even_2_out_ga₁(B)) -> even_2_out_ga₁(B)
half_2_in_ga₁(0_0) -> half_2_out_ga₁(0_0)
even_2_in_ga₁(0_0) -> even_2_out_ga₁(true_0)

The set Q consists of the following terms:

half_2_in_ga₁(x0)
even_2_in_ga₁(x0)
if_half_2_in_1_ga₁(x0)
if_even_2_in_1_ga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_IF_4_IN_1_GGGA₂, IF_4_IN_GGGA₃, TIMES_3_IN_GGA₂, IF_TIMES_3_IN_1_GGA₃}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

TIMES_3_IN_GGA₂(s_1₁(X), Y) -> IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_in_ga₁(s_1₁(X)))

The remaining Dependency Pairs were at least non-strictly be oriented.

IF_4_IN_GGGA₃(true_0, s_1₁(X), Y) -> IF_IF_4_IN_1_GGGA₂(Y, half_2_in_ga₁(s_1₁(X)))
IF_IF_4_IN_1_GGGA₂(Y, half_2_out_ga₁(X1)) -> TIMES_3_IN_GGA₂(X1, Y)
IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_out_ga₁(B)) -> IF_4_IN_GGGA₃(B, s_1₁(X), Y)

With the implicit AFS we had to orient the following set of usable rules non-strictly.

if_half_2_in_1_ga₁(half_2_out_ga₁(Y)) -> half_2_out_ga₁(s_1₁(Y))
half_2_in_ga₁(0_0) -> half_2_out_ga₁(0_0)
half_2_in_ga₁(s_1₁(s_1₁(X))) -> if_half_2_in_1_ga₁(half_2_in_ga₁(X))

Used ordering: POLO with Polynomial interpretation:

POL(0_0) = 0
POL(if_half_2_in_1_ga₁(x₁)) = 2 + x₁
POL(TIMES_3_IN_GGA₂(x₁, x₂)) = 2·x₁ + x₂
POL(false_0) = 0
POL(half_2_in_ga₁(x₁)) = x₁
POL(true_0) = 0
POL(IF_TIMES_3_IN_1_GGA₃(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(half_2_out_ga₁(x₁)) = 2·x₁
POL(IF_4_IN_GGGA₃(x₁, x₂, x₃)) = x₂ + x₃
POL(if_even_2_in_1_ga₁(x₁)) = 0
POL(even_2_in_ga₁(x₁)) = 0
POL(even_2_out_ga₁(x₁)) = 0
POL(IF_IF_4_IN_1_GGGA₂(x₁, x₂)) = x₁ + x₂
POL(s_1₁(x₁)) = 1 + x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

                          ↳ QDP

                            ↳ QDPPoloProof

                              ↳ QDP

                                ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

IF_4_IN_GGGA₃(true_0, s_1₁(X), Y) -> IF_IF_4_IN_1_GGGA₂(Y, half_2_in_ga₁(s_1₁(X)))
IF_IF_4_IN_1_GGGA₂(Y, half_2_out_ga₁(X1)) -> TIMES_3_IN_GGA₂(X1, Y)
IF_TIMES_3_IN_1_GGA₃(X, Y, even_2_out_ga₁(B)) -> IF_4_IN_GGGA₃(B, s_1₁(X), Y)

The TRS R consists of the following rules:

half_2_in_ga₁(s_1₁(s_1₁(X))) -> if_half_2_in_1_ga₁(half_2_in_ga₁(X))
even_2_in_ga₁(s_1₁(0_0)) -> even_2_out_ga₁(false_0)
even_2_in_ga₁(s_1₁(s_1₁(X))) -> if_even_2_in_1_ga₁(even_2_in_ga₁(X))
if_half_2_in_1_ga₁(half_2_out_ga₁(Y)) -> half_2_out_ga₁(s_1₁(Y))
if_even_2_in_1_ga₁(even_2_out_ga₁(B)) -> even_2_out_ga₁(B)
half_2_in_ga₁(0_0) -> half_2_out_ga₁(0_0)
even_2_in_ga₁(0_0) -> even_2_out_ga₁(true_0)

The set Q consists of the following terms:

half_2_in_ga₁(x0)
even_2_in_ga₁(x0)
if_half_2_in_1_ga₁(x0)
if_even_2_in_1_ga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_IF_4_IN_1_GGGA₂, IF_4_IN_GGGA₃, TIMES_3_IN_GGA₂, IF_TIMES_3_IN_1_GGA₃}.
The approximation of the Dependency Graph contains 0 SCCs with 3 less nodes.
With regard to the inferred argument filtering the predicates were used in the following modes:
times₃: (b,b,f)
even₂: (b,f)
if₄: (b,b,b,f)
half₂: (b,f)
plus₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

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

times_3_in_gga₃(0_0, Y, 0_0) -> times_3_out_gga₃(0_0, Y, 0_0)
times_3_in_gga₃(s_1₁(X), Y, Z) -> if_times_3_in_1_gga₄(X, Y, Z, even_2_in_ga₂(s_1₁(X), B))
even_2_in_ga₂(0_0, true_0) -> even_2_out_ga₂(0_0, true_0)
even_2_in_ga₂(s_1₁(0_0), false_0) -> even_2_out_ga₂(s_1₁(0_0), false_0)
even_2_in_ga₂(s_1₁(s_1₁(X)), B) -> if_even_2_in_1_ga₃(X, B, even_2_in_ga₂(X, B))
if_even_2_in_1_ga₃(X, B, even_2_out_ga₂(X, B)) -> even_2_out_ga₂(s_1₁(s_1₁(X)), B)
if_times_3_in_1_gga₄(X, Y, Z, even_2_out_ga₂(s_1₁(X), B)) -> if_times_3_in_2_gga₅(X, Y, Z, B, if_4_in_ggga₄(B, s_1₁(X), Y, Z))
if_4_in_ggga₄(true_0, s_1₁(X), Y, Z) -> if_if_4_in_1_ggga₄(X, Y, Z, half_2_in_ga₂(s_1₁(X), X1))
half_2_in_ga₂(0_0, 0_0) -> half_2_out_ga₂(0_0, 0_0)
half_2_in_ga₂(s_1₁(s_1₁(X)), s_1₁(Y)) -> if_half_2_in_1_ga₃(X, Y, half_2_in_ga₂(X, Y))
if_half_2_in_1_ga₃(X, Y, half_2_out_ga₂(X, Y)) -> half_2_out_ga₂(s_1₁(s_1₁(X)), s_1₁(Y))
if_if_4_in_1_ggga₄(X, Y, Z, half_2_out_ga₂(s_1₁(X), X1)) -> if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_in_gga₃(X1, Y, Y1))
if_if_4_in_2_ggga₅(X, Y, Z, X1, times_3_out_gga₃(X1, Y, Y1)) -> if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_in_gga₃(Y1, Y1, Z))
plus_3_in_gga₃(0_0, Y, Y) -> plus_3_out_gga₃(0_0, Y, Y)
plus_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_plus_3_in_1_gga₄(X, Y, Z, plus_3_in_gga₃(X, Y, Z))
if_plus_3_in_1_gga₄(X, Y, Z, plus_3_out_gga₃(X, Y, Z)) -> plus_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_if_4_in_3_ggga₅(X, Y, Z, Y1, plus_3_out_gga₃(Y1, Y1, Z)) -> if_4_out_ggga₄(true_0, s_1₁(X), Y, Z)
if_4_in_ggga₄(false_0, s_1₁(X), Y, Z) -> if_if_4_in_4_ggga₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
if_if_4_in_4_ggga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_in_gga₃(Y, U, Z))
if_if_4_in_5_ggga₅(X, Y, Z, U, plus_3_out_gga₃(Y, U, Z)) -> if_4_out_ggga₄(false_0, s_1₁(X), Y, Z)
if_times_3_in_2_gga₅(X, Y, Z, B, if_4_out_ggga₄(B, s_1₁(X), Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)