Left Termination proof of /tmp/tmpIL10ZA/prime.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

div₃(X, Y, Z) :- quot₄(X, Y, Y, Z).
quot₄(0₀, s₁(Y), s₁(Z), 0₀).
quot₄(s₁(X), s₁(Y), Z, U) :- quot₄(X, Y, Z, U).
quot₄(X, 0₀, s₁(Z), s₁(U)) :- quot₄(X, s₁(Z), s₁(Z), U).
prime₁(s₁²(X)) :- pr₂(s₁²(X), s₁(X)).
pr₂(X, s₁(0₀)).
pr₂(X, s₁²(Y)) :- not_divides₂(s₁²(Y), X), pr₂(X, s₁(Y)).
not_divides₂(Y, X) :- div₃(X, Y, U), times₃(U, Y, Z), neq₂(X, Z).
neq₂(s₁(X), 0₀).
neq₂(0₀, s₁(X)).
neq₂(s₁(X), s₁(Y)) :- neq₂(X, Y).
times₃(0₀, Y, 0₀).
times₃(s₁(X), Y, Z) :- times₃(X, Y, U), add₃(U, Y, Z).
add₃(X, 0₀, X).
add₃(0₀, X, X).
add₃(s₁(X), Y, s₁(Z)) :- add₃(X, Y, Z).

With regard to the inferred argument filtering the predicates were used in the following modes:
prime₁: (b)
pr₂: (b,b)
not_divides₂: (b,b)
div₃: (b,b,f)
quot₄: (b,b,b,f)
times₃: (b,b,f)
add₃: (b,b,f)
neq₂: (b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

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:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

PRIME_1_IN_G₁(s_1₁(s_1₁(X))) -> IF_PRIME_1_IN_1_G₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
PRIME_1_IN_G₁(s_1₁(s_1₁(X))) -> PR_2_IN_GG₂(s_1₁(s_1₁(X)), s_1₁(X))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> NOT_DIVIDES_2_IN_GG₂(s_1₁(s_1₁(Y)), X)
NOT_DIVIDES_2_IN_GG₂(Y, X) -> IF_NOT_DIVIDES_2_IN_1_GG₃(Y, X, div_3_in_gga₃(X, Y, U))
NOT_DIVIDES_2_IN_GG₂(Y, X) -> DIV_3_IN_GGA₃(X, Y, U)
DIV_3_IN_GGA₃(X, Y, Z) -> IF_DIV_3_IN_1_GGA₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
DIV_3_IN_GGA₃(X, Y, Z) -> QUOT_4_IN_GGGA₄(X, Y, Y, Z)
QUOT_4_IN_GGGA₄(s_1₁(X), s_1₁(Y), Z, U) -> IF_QUOT_4_IN_1_GGGA₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
QUOT_4_IN_GGGA₄(s_1₁(X), s_1₁(Y), Z, U) -> QUOT_4_IN_GGGA₄(X, Y, Z, U)
QUOT_4_IN_GGGA₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> IF_QUOT_4_IN_2_GGGA₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
QUOT_4_IN_GGGA₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> QUOT_4_IN_GGGA₄(X, s_1₁(Z), s_1₁(Z), U)
IF_NOT_DIVIDES_2_IN_1_GG₃(Y, X, div_3_out_gga₃(X, Y, U)) -> IF_NOT_DIVIDES_2_IN_2_GG₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
IF_NOT_DIVIDES_2_IN_1_GG₃(Y, X, div_3_out_gga₃(X, Y, U)) -> TIMES_3_IN_GGA₃(U, Y, Z)
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> IF_TIMES_3_IN_2_GGA₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> ADD_3_IN_GGA₃(U, Y, Z)
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GGA₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)
IF_NOT_DIVIDES_2_IN_2_GG₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> IF_NOT_DIVIDES_2_IN_3_GG₄(Y, X, Z, neq_2_in_gg₂(X, Z))
IF_NOT_DIVIDES_2_IN_2_GG₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> NEQ_2_IN_GG₂(X, Z)
NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_NEQ_2_IN_1_GG₃(X, Y, neq_2_in_gg₂(X, Y))
NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NEQ_2_IN_GG₂(X, Y)
IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> IF_PR_2_IN_2_GG₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> PR_2_IN_GG₂(X, s_1₁(Y))

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

The argument filtering Pi contains the following mapping:
prime_1_in_g₁(x1) = prime_1_in_g₁(x1)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
if_prime_1_in_1_g₂(x1, x2) = if_prime_1_in_1_g₁(x2)
pr_2_in_gg₂(x1, x2) = pr_2_in_gg₂(x1, x2)
pr_2_out_gg₂(x1, x2) = pr_2_out_gg
if_pr_2_in_1_gg₃(x1, x2, x3) = if_pr_2_in_1_gg₃(x1, x2, x3)
not_divides_2_in_gg₂(x1, x2) = not_divides_2_in_gg₂(x1, x2)
if_not_divides_2_in_1_gg₃(x1, x2, x3) = if_not_divides_2_in_1_gg₃(x1, x2, x3)
div_3_in_gga₃(x1, x2, x3) = div_3_in_gga₂(x1, x2)
if_div_3_in_1_gga₄(x1, x2, x3, x4) = if_div_3_in_1_gga₁(x4)
quot_4_in_ggga₄(x1, x2, x3, x4) = quot_4_in_ggga₃(x1, x2, x3)
quot_4_out_ggga₄(x1, x2, x3, x4) = quot_4_out_ggga₁(x4)
if_quot_4_in_1_ggga₅(x1, x2, x3, x4, x5) = if_quot_4_in_1_ggga₁(x5)
if_quot_4_in_2_ggga₄(x1, x2, x3, x4) = if_quot_4_in_2_ggga₁(x4)
div_3_out_gga₃(x1, x2, x3) = div_3_out_gga₁(x3)
if_not_divides_2_in_2_gg₄(x1, x2, x3, x4) = if_not_divides_2_in_2_gg₂(x2, x4)
times_3_in_gga₃(x1, x2, x3) = times_3_in_gga₂(x1, x2)
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_gga₂(x2, x4)
if_times_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_times_3_in_2_gga₁(x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
if_not_divides_2_in_3_gg₄(x1, x2, x3, x4) = if_not_divides_2_in_3_gg₁(x4)
neq_2_in_gg₂(x1, x2) = neq_2_in_gg₂(x1, x2)
neq_2_out_gg₂(x1, x2) = neq_2_out_gg
if_neq_2_in_1_gg₃(x1, x2, x3) = if_neq_2_in_1_gg₁(x3)
not_divides_2_out_gg₂(x1, x2) = not_divides_2_out_gg
if_pr_2_in_2_gg₃(x1, x2, x3) = if_pr_2_in_2_gg₁(x3)
prime_1_out_g₁(x1) = prime_1_out_g
IF_TIMES_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_GGA₂(x2, x4)
QUOT_4_IN_GGGA₄(x1, x2, x3, x4) = QUOT_4_IN_GGGA₃(x1, x2, x3)
PR_2_IN_GG₂(x1, x2) = PR_2_IN_GG₂(x1, x2)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_3_IN_GGA₂(x1, x2)
NOT_DIVIDES_2_IN_GG₂(x1, x2) = NOT_DIVIDES_2_IN_GG₂(x1, x2)
TIMES_3_IN_GGA₃(x1, x2, x3) = TIMES_3_IN_GGA₂(x1, x2)
IF_NEQ_2_IN_1_GG₃(x1, x2, x3) = IF_NEQ_2_IN_1_GG₁(x3)
IF_QUOT_4_IN_1_GGGA₅(x1, x2, x3, x4, x5) = IF_QUOT_4_IN_1_GGGA₁(x5)
IF_NOT_DIVIDES_2_IN_3_GG₄(x1, x2, x3, x4) = IF_NOT_DIVIDES_2_IN_3_GG₁(x4)
IF_PR_2_IN_1_GG₃(x1, x2, x3) = IF_PR_2_IN_1_GG₃(x1, x2, x3)
IF_PR_2_IN_2_GG₃(x1, x2, x3) = IF_PR_2_IN_2_GG₁(x3)
NEQ_2_IN_GG₂(x1, x2) = NEQ_2_IN_GG₂(x1, x2)
DIV_3_IN_GGA₃(x1, x2, x3) = DIV_3_IN_GGA₂(x1, x2)
IF_TIMES_3_IN_2_GGA₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_2_GGA₁(x5)
PRIME_1_IN_G₁(x1) = PRIME_1_IN_G₁(x1)
IF_NOT_DIVIDES_2_IN_1_GG₃(x1, x2, x3) = IF_NOT_DIVIDES_2_IN_1_GG₃(x1, x2, x3)
IF_ADD_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_ADD_3_IN_1_GGA₁(x4)
IF_NOT_DIVIDES_2_IN_2_GG₄(x1, x2, x3, x4) = IF_NOT_DIVIDES_2_IN_2_GG₂(x2, x4)
IF_DIV_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_DIV_3_IN_1_GGA₁(x4)
IF_QUOT_4_IN_2_GGGA₄(x1, x2, x3, x4) = IF_QUOT_4_IN_2_GGGA₁(x4)
IF_PRIME_1_IN_1_G₂(x1, x2) = IF_PRIME_1_IN_1_G₁(x2)

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:

PRIME_1_IN_G₁(s_1₁(s_1₁(X))) -> IF_PRIME_1_IN_1_G₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
PRIME_1_IN_G₁(s_1₁(s_1₁(X))) -> PR_2_IN_GG₂(s_1₁(s_1₁(X)), s_1₁(X))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> NOT_DIVIDES_2_IN_GG₂(s_1₁(s_1₁(Y)), X)
NOT_DIVIDES_2_IN_GG₂(Y, X) -> IF_NOT_DIVIDES_2_IN_1_GG₃(Y, X, div_3_in_gga₃(X, Y, U))
NOT_DIVIDES_2_IN_GG₂(Y, X) -> DIV_3_IN_GGA₃(X, Y, U)
DIV_3_IN_GGA₃(X, Y, Z) -> IF_DIV_3_IN_1_GGA₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
DIV_3_IN_GGA₃(X, Y, Z) -> QUOT_4_IN_GGGA₄(X, Y, Y, Z)
QUOT_4_IN_GGGA₄(s_1₁(X), s_1₁(Y), Z, U) -> IF_QUOT_4_IN_1_GGGA₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
QUOT_4_IN_GGGA₄(s_1₁(X), s_1₁(Y), Z, U) -> QUOT_4_IN_GGGA₄(X, Y, Z, U)
QUOT_4_IN_GGGA₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> IF_QUOT_4_IN_2_GGGA₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
QUOT_4_IN_GGGA₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> QUOT_4_IN_GGGA₄(X, s_1₁(Z), s_1₁(Z), U)
IF_NOT_DIVIDES_2_IN_1_GG₃(Y, X, div_3_out_gga₃(X, Y, U)) -> IF_NOT_DIVIDES_2_IN_2_GG₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
IF_NOT_DIVIDES_2_IN_1_GG₃(Y, X, div_3_out_gga₃(X, Y, U)) -> TIMES_3_IN_GGA₃(U, Y, Z)
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> IF_TIMES_3_IN_1_GGA₄(X, Y, Z, times_3_in_gga₃(X, Y, U))
TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> IF_TIMES_3_IN_2_GGA₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
IF_TIMES_3_IN_1_GGA₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> ADD_3_IN_GGA₃(U, Y, Z)
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GGA₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)
IF_NOT_DIVIDES_2_IN_2_GG₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> IF_NOT_DIVIDES_2_IN_3_GG₄(Y, X, Z, neq_2_in_gg₂(X, Z))
IF_NOT_DIVIDES_2_IN_2_GG₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> NEQ_2_IN_GG₂(X, Z)
NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_NEQ_2_IN_1_GG₃(X, Y, neq_2_in_gg₂(X, Y))
NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NEQ_2_IN_GG₂(X, Y)
IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> IF_PR_2_IN_2_GG₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> PR_2_IN_GG₂(X, s_1₁(Y))

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NEQ_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NEQ_2_IN_GG₂(X, Y)

R is empty.
Pi is empty.
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

              ↳ PiDP

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

NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NEQ_2_IN_GG₂(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 {NEQ_2_IN_GG₂}.
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:

NEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NEQ_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

ADD_3_IN_GGA₂(s_1₁(X), Y) -> ADD_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 {ADD_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:

ADD_3_IN_GGA₂(s_1₁(X), Y) -> ADD_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

TIMES_3_IN_GGA₃(s_1₁(X), Y, Z) -> TIMES_3_IN_GGA₃(X, Y, U)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
TIMES_3_IN_GGA₃(x1, x2, x3) = TIMES_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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

TIMES_3_IN_GGA₂(s_1₁(X), Y) -> TIMES_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 {TIMES_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:

TIMES_3_IN_GGA₂(s_1₁(X), Y) -> TIMES_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

QUOT_4_IN_GGGA₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> QUOT_4_IN_GGGA₄(X, s_1₁(Z), s_1₁(Z), U)
QUOT_4_IN_GGGA₄(s_1₁(X), s_1₁(Y), Z, U) -> QUOT_4_IN_GGGA₄(X, Y, Z, U)

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

QUOT_4_IN_GGGA₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> QUOT_4_IN_GGGA₄(X, s_1₁(Z), s_1₁(Z), U)
QUOT_4_IN_GGGA₄(s_1₁(X), s_1₁(Y), Z, U) -> QUOT_4_IN_GGGA₄(X, Y, Z, U)

R is empty.
The argument filtering Pi contains the following mapping:
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
QUOT_4_IN_GGGA₄(x1, x2, x3, x4) = QUOT_4_IN_GGGA₃(x1, x2, x3)

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

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

QUOT_4_IN_GGGA₃(X, 0_0, s_1₁(Z)) -> QUOT_4_IN_GGGA₃(X, s_1₁(Z), s_1₁(Z))
QUOT_4_IN_GGGA₃(s_1₁(X), s_1₁(Y), Z) -> QUOT_4_IN_GGGA₃(X, Y, Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {QUOT_4_IN_GGGA₃}.
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:

QUOT_4_IN_GGGA₃(s_1₁(X), s_1₁(Y), Z) -> QUOT_4_IN_GGGA₃(X, Y, Z)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
QUOT_4_IN_GGGA₃(X, 0_0, s_1₁(Z)) -> QUOT_4_IN_GGGA₃(X, s_1₁(Z), s_1₁(Z))
The graph contains the following edges 1 >= 1, 3 >= 2, 3 >= 3

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> PR_2_IN_GG₂(X, s_1₁(Y))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))

The TRS R consists of the following rules:

prime_1_in_g₁(s_1₁(s_1₁(X))) -> if_prime_1_in_1_g₂(X, pr_2_in_gg₂(s_1₁(s_1₁(X)), s_1₁(X)))
pr_2_in_gg₂(X, s_1₁(0_0)) -> pr_2_out_gg₂(X, s_1₁(0_0))
pr_2_in_gg₂(X, s_1₁(s_1₁(Y))) -> if_pr_2_in_1_gg₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))
not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
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, times_3_in_gga₃(X, Y, U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
if_pr_2_in_1_gg₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> if_pr_2_in_2_gg₃(X, Y, pr_2_in_gg₂(X, s_1₁(Y)))
if_pr_2_in_2_gg₃(X, Y, pr_2_out_gg₂(X, s_1₁(Y))) -> pr_2_out_gg₂(X, s_1₁(s_1₁(Y)))
if_prime_1_in_1_g₂(X, pr_2_out_gg₂(s_1₁(s_1₁(X)), s_1₁(X))) -> prime_1_out_g₁(s_1₁(s_1₁(X)))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg₂(s_1₁(s_1₁(Y)), X)) -> PR_2_IN_GG₂(X, s_1₁(Y))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))

The TRS R consists of the following rules:

not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₃(X, Y, U))
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₃(X, Y, U)) -> if_not_divides_2_in_2_gg₄(Y, X, U, times_3_in_gga₃(U, Y, Z))
div_3_in_gga₃(X, Y, Z) -> if_div_3_in_1_gga₄(X, Y, Z, quot_4_in_ggga₄(X, Y, Y, Z))
if_not_divides_2_in_2_gg₄(Y, X, U, times_3_out_gga₃(U, Y, Z)) -> if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_in_gg₂(X, Z))
if_div_3_in_1_gga₄(X, Y, Z, quot_4_out_ggga₄(X, Y, Y, Z)) -> div_3_out_gga₃(X, Y, Z)
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, times_3_in_gga₃(X, Y, U))
if_not_divides_2_in_3_gg₄(Y, X, Z, neq_2_out_gg₂(X, Z)) -> not_divides_2_out_gg₂(Y, X)
quot_4_in_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0) -> quot_4_out_ggga₄(0_0, s_1₁(Y), s_1₁(Z), 0_0)
quot_4_in_ggga₄(s_1₁(X), s_1₁(Y), Z, U) -> if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_in_ggga₄(X, Y, Z, U))
quot_4_in_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U)) -> if_quot_4_in_2_ggga₄(X, Z, U, quot_4_in_ggga₄(X, s_1₁(Z), s_1₁(Z), U))
if_times_3_in_1_gga₄(X, Y, Z, times_3_out_gga₃(X, Y, U)) -> if_times_3_in_2_gga₅(X, Y, Z, U, add_3_in_gga₃(U, Y, Z))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg₂(s_1₁(X), 0_0)
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg₂(0_0, s_1₁(X))
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₃(X, Y, neq_2_in_gg₂(X, Y))
if_quot_4_in_1_ggga₅(X, Y, Z, U, quot_4_out_ggga₄(X, Y, Z, U)) -> quot_4_out_ggga₄(s_1₁(X), s_1₁(Y), Z, U)
if_quot_4_in_2_ggga₄(X, Z, U, quot_4_out_ggga₄(X, s_1₁(Z), s_1₁(Z), U)) -> quot_4_out_ggga₄(X, 0_0, s_1₁(Z), s_1₁(U))
if_times_3_in_2_gga₅(X, Y, Z, U, add_3_out_gga₃(U, Y, Z)) -> times_3_out_gga₃(s_1₁(X), Y, Z)
if_neq_2_in_1_gg₃(X, Y, neq_2_out_gg₂(X, Y)) -> neq_2_out_gg₂(s_1₁(X), s_1₁(Y))
add_3_in_gga₃(X, 0_0, X) -> add_3_out_gga₃(X, 0_0, X)
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))

The argument filtering Pi contains the following mapping:
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
not_divides_2_in_gg₂(x1, x2) = not_divides_2_in_gg₂(x1, x2)
if_not_divides_2_in_1_gg₃(x1, x2, x3) = if_not_divides_2_in_1_gg₃(x1, x2, x3)
div_3_in_gga₃(x1, x2, x3) = div_3_in_gga₂(x1, x2)
if_div_3_in_1_gga₄(x1, x2, x3, x4) = if_div_3_in_1_gga₁(x4)
quot_4_in_ggga₄(x1, x2, x3, x4) = quot_4_in_ggga₃(x1, x2, x3)
quot_4_out_ggga₄(x1, x2, x3, x4) = quot_4_out_ggga₁(x4)
if_quot_4_in_1_ggga₅(x1, x2, x3, x4, x5) = if_quot_4_in_1_ggga₁(x5)
if_quot_4_in_2_ggga₄(x1, x2, x3, x4) = if_quot_4_in_2_ggga₁(x4)
div_3_out_gga₃(x1, x2, x3) = div_3_out_gga₁(x3)
if_not_divides_2_in_2_gg₄(x1, x2, x3, x4) = if_not_divides_2_in_2_gg₂(x2, x4)
times_3_in_gga₃(x1, x2, x3) = times_3_in_gga₂(x1, x2)
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_gga₂(x2, x4)
if_times_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_times_3_in_2_gga₁(x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
if_not_divides_2_in_3_gg₄(x1, x2, x3, x4) = if_not_divides_2_in_3_gg₁(x4)
neq_2_in_gg₂(x1, x2) = neq_2_in_gg₂(x1, x2)
neq_2_out_gg₂(x1, x2) = neq_2_out_gg
if_neq_2_in_1_gg₃(x1, x2, x3) = if_neq_2_in_1_gg₁(x3)
not_divides_2_out_gg₂(x1, x2) = not_divides_2_out_gg
PR_2_IN_GG₂(x1, x2) = PR_2_IN_GG₂(x1, x2)
IF_PR_2_IN_1_GG₃(x1, x2, x3) = IF_PR_2_IN_1_GG₃(x1, x2, x3)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

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

IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg) -> PR_2_IN_GG₂(X, s_1₁(Y))
PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))

The TRS R consists of the following rules:

not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₂(X, Y))
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₁(U)) -> if_not_divides_2_in_2_gg₂(X, times_3_in_gga₂(U, Y))
div_3_in_gga₂(X, Y) -> if_div_3_in_1_gga₁(quot_4_in_ggga₃(X, Y, Y))
if_not_divides_2_in_2_gg₂(X, times_3_out_gga₁(Z)) -> if_not_divides_2_in_3_gg₁(neq_2_in_gg₂(X, Z))
if_div_3_in_1_gga₁(quot_4_out_ggga₁(Z)) -> div_3_out_gga₁(Z)
times_3_in_gga₂(0_0, Y) -> times_3_out_gga₁(0_0)
times_3_in_gga₂(s_1₁(X), Y) -> if_times_3_in_1_gga₂(Y, times_3_in_gga₂(X, Y))
if_not_divides_2_in_3_gg₁(neq_2_out_gg) -> not_divides_2_out_gg
quot_4_in_ggga₃(0_0, s_1₁(Y), s_1₁(Z)) -> quot_4_out_ggga₁(0_0)
quot_4_in_ggga₃(s_1₁(X), s_1₁(Y), Z) -> if_quot_4_in_1_ggga₁(quot_4_in_ggga₃(X, Y, Z))
quot_4_in_ggga₃(X, 0_0, s_1₁(Z)) -> if_quot_4_in_2_ggga₁(quot_4_in_ggga₃(X, s_1₁(Z), s_1₁(Z)))
if_times_3_in_1_gga₂(Y, times_3_out_gga₁(U)) -> if_times_3_in_2_gga₁(add_3_in_gga₂(U, Y))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₁(neq_2_in_gg₂(X, Y))
if_quot_4_in_1_ggga₁(quot_4_out_ggga₁(U)) -> quot_4_out_ggga₁(U)
if_quot_4_in_2_ggga₁(quot_4_out_ggga₁(U)) -> quot_4_out_ggga₁(s_1₁(U))
if_times_3_in_2_gga₁(add_3_out_gga₁(Z)) -> times_3_out_gga₁(Z)
if_neq_2_in_1_gg₁(neq_2_out_gg) -> neq_2_out_gg
add_3_in_gga₂(X, 0_0) -> add_3_out_gga₁(X)
add_3_in_gga₂(0_0, X) -> add_3_out_gga₁(X)
add_3_in_gga₂(s_1₁(X), Y) -> if_add_3_in_1_gga₁(add_3_in_gga₂(X, Y))
if_add_3_in_1_gga₁(add_3_out_gga₁(Z)) -> add_3_out_gga₁(s_1₁(Z))

The set Q consists of the following terms:

not_divides_2_in_gg₂(x0, x1)
if_not_divides_2_in_1_gg₃(x0, x1, x2)
div_3_in_gga₂(x0, x1)
if_not_divides_2_in_2_gg₂(x0, x1)
if_div_3_in_1_gga₁(x0)
times_3_in_gga₂(x0, x1)
if_not_divides_2_in_3_gg₁(x0)
quot_4_in_ggga₃(x0, x1, x2)
if_times_3_in_1_gga₂(x0, x1)
neq_2_in_gg₂(x0, x1)
if_quot_4_in_1_ggga₁(x0)
if_quot_4_in_2_ggga₁(x0)
if_times_3_in_2_gga₁(x0)
if_neq_2_in_1_gg₁(x0)
add_3_in_gga₂(x0, x1)
if_add_3_in_1_gga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {PR_2_IN_GG₂, IF_PR_2_IN_1_GG₃}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

PR_2_IN_GG₂(X, s_1₁(s_1₁(Y))) -> IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_in_gg₂(s_1₁(s_1₁(Y)), X))

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

IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg) -> PR_2_IN_GG₂(X, s_1₁(Y))

With the implicit AFS there is no usable rule.

Used ordering: POLO with Polynomial interpretation:

POL(0_0) = 0
POL(if_quot_4_in_1_ggga₁(x₁)) = 0
POL(div_3_in_gga₂(x₁, x₂)) = 0
POL(not_divides_2_in_gg₂(x₁, x₂)) = 0
POL(not_divides_2_out_gg) = 0
POL(if_not_divides_2_in_2_gg₂(x₁, x₂)) = 0
POL(neq_2_in_gg₂(x₁, x₂)) = 0
POL(add_3_in_gga₂(x₁, x₂)) = 0
POL(if_div_3_in_1_gga₁(x₁)) = 0
POL(div_3_out_gga₁(x₁)) = 0
POL(if_neq_2_in_1_gg₁(x₁)) = 0
POL(PR_2_IN_GG₂(x₁, x₂)) = x₂
POL(if_times_3_in_1_gga₂(x₁, x₂)) = 0
POL(if_add_3_in_1_gga₁(x₁)) = 0
POL(add_3_out_gga₁(x₁)) = 0
POL(IF_PR_2_IN_1_GG₃(x₁, x₂, x₃)) = 1 + x₂
POL(quot_4_in_ggga₃(x₁, x₂, x₃)) = 0
POL(quot_4_out_ggga₁(x₁)) = 0
POL(times_3_out_gga₁(x₁)) = 0
POL(if_not_divides_2_in_1_gg₃(x₁, x₂, x₃)) = 0
POL(if_quot_4_in_2_ggga₁(x₁)) = 0
POL(if_times_3_in_2_gga₁(x₁)) = 0
POL(times_3_in_gga₂(x₁, x₂)) = 0
POL(s_1₁(x₁)) = 1 + x₁
POL(neq_2_out_gg) = 0
POL(if_not_divides_2_in_3_gg₁(x₁)) = 0

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

IF_PR_2_IN_1_GG₃(X, Y, not_divides_2_out_gg) -> PR_2_IN_GG₂(X, s_1₁(Y))

The TRS R consists of the following rules:

not_divides_2_in_gg₂(Y, X) -> if_not_divides_2_in_1_gg₃(Y, X, div_3_in_gga₂(X, Y))
if_not_divides_2_in_1_gg₃(Y, X, div_3_out_gga₁(U)) -> if_not_divides_2_in_2_gg₂(X, times_3_in_gga₂(U, Y))
div_3_in_gga₂(X, Y) -> if_div_3_in_1_gga₁(quot_4_in_ggga₃(X, Y, Y))
if_not_divides_2_in_2_gg₂(X, times_3_out_gga₁(Z)) -> if_not_divides_2_in_3_gg₁(neq_2_in_gg₂(X, Z))
if_div_3_in_1_gga₁(quot_4_out_ggga₁(Z)) -> div_3_out_gga₁(Z)
times_3_in_gga₂(0_0, Y) -> times_3_out_gga₁(0_0)
times_3_in_gga₂(s_1₁(X), Y) -> if_times_3_in_1_gga₂(Y, times_3_in_gga₂(X, Y))
if_not_divides_2_in_3_gg₁(neq_2_out_gg) -> not_divides_2_out_gg
quot_4_in_ggga₃(0_0, s_1₁(Y), s_1₁(Z)) -> quot_4_out_ggga₁(0_0)
quot_4_in_ggga₃(s_1₁(X), s_1₁(Y), Z) -> if_quot_4_in_1_ggga₁(quot_4_in_ggga₃(X, Y, Z))
quot_4_in_ggga₃(X, 0_0, s_1₁(Z)) -> if_quot_4_in_2_ggga₁(quot_4_in_ggga₃(X, s_1₁(Z), s_1₁(Z)))
if_times_3_in_1_gga₂(Y, times_3_out_gga₁(U)) -> if_times_3_in_2_gga₁(add_3_in_gga₂(U, Y))
neq_2_in_gg₂(s_1₁(X), 0_0) -> neq_2_out_gg
neq_2_in_gg₂(0_0, s_1₁(X)) -> neq_2_out_gg
neq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_neq_2_in_1_gg₁(neq_2_in_gg₂(X, Y))
if_quot_4_in_1_ggga₁(quot_4_out_ggga₁(U)) -> quot_4_out_ggga₁(U)
if_quot_4_in_2_ggga₁(quot_4_out_ggga₁(U)) -> quot_4_out_ggga₁(s_1₁(U))
if_times_3_in_2_gga₁(add_3_out_gga₁(Z)) -> times_3_out_gga₁(Z)
if_neq_2_in_1_gg₁(neq_2_out_gg) -> neq_2_out_gg
add_3_in_gga₂(X, 0_0) -> add_3_out_gga₁(X)
add_3_in_gga₂(0_0, X) -> add_3_out_gga₁(X)
add_3_in_gga₂(s_1₁(X), Y) -> if_add_3_in_1_gga₁(add_3_in_gga₂(X, Y))
if_add_3_in_1_gga₁(add_3_out_gga₁(Z)) -> add_3_out_gga₁(s_1₁(Z))

The set Q consists of the following terms:

not_divides_2_in_gg₂(x0, x1)
if_not_divides_2_in_1_gg₃(x0, x1, x2)
div_3_in_gga₂(x0, x1)
if_not_divides_2_in_2_gg₂(x0, x1)
if_div_3_in_1_gga₁(x0)
times_3_in_gga₂(x0, x1)
if_not_divides_2_in_3_gg₁(x0)
quot_4_in_ggga₃(x0, x1, x2)
if_times_3_in_1_gga₂(x0, x1)
neq_2_in_gg₂(x0, x1)
if_quot_4_in_1_ggga₁(x0)
if_quot_4_in_2_ggga₁(x0)
if_times_3_in_2_gga₁(x0)
if_neq_2_in_1_gg₁(x0)
add_3_in_gga₂(x0, x1)
if_add_3_in_1_gga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {PR_2_IN_GG₂, IF_PR_2_IN_1_GG₃}.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.