Left Termination of the query pattern rem_in_3(g, a, g) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

rem(X, Y, R) :- ','(notZero(Y), ','(sub(X, Y, Z), rem(Z, Y, R))).
rem(X, Y, X) :- ','(notZero(Y), geq(X, Y)).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).
notZero(s(X)).
geq(s(X), s(Y)) :- geq(X, Y).
geq(X, 0).

Queries:

rem(g,a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
rem_in: (b,f,b) (b,b,b)
sub_in: (b,f,f) (b,b,f)
geq_in: (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)

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:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

REM_IN_GAG(X, Y, R) → U1_GAG(X, Y, R, notZero_in_a(Y))
REM_IN_GAG(X, Y, R) → NOTZERO_IN_A(Y)
U1_GAG(X, Y, R, notZero_out_a(Y)) → U2_GAG(X, Y, R, sub_in_gaa(X, Y, Z))
U1_GAG(X, Y, R, notZero_out_a(Y)) → SUB_IN_GAA(X, Y, Z)
SUB_IN_GAA(s(X), s(Y), Z) → U6_GAA(X, Y, Z, sub_in_gaa(X, Y, Z))
SUB_IN_GAA(s(X), s(Y), Z) → SUB_IN_GAA(X, Y, Z)
U2_GAG(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_GAG(X, Y, R, rem_in_ggg(Z, Y, R))
U2_GAG(X, Y, R, sub_out_gaa(X, Y, Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y))
REM_IN_GGG(X, Y, R) → NOTZERO_IN_G(Y)
U1_GGG(X, Y, R, notZero_out_g(Y)) → U2_GGG(X, Y, R, sub_in_gga(X, Y, Z))
U1_GGG(X, Y, R, notZero_out_g(Y)) → SUB_IN_GGA(X, Y, Z)
SUB_IN_GGA(s(X), s(Y), Z) → U6_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)
U2_GGG(X, Y, R, sub_out_gga(X, Y, Z)) → U3_GGG(X, Y, R, rem_in_ggg(Z, Y, R))
U2_GGG(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, X) → U4_GGG(X, Y, notZero_in_g(Y))
REM_IN_GGG(X, Y, X) → NOTZERO_IN_G(Y)
U4_GGG(X, Y, notZero_out_g(Y)) → U5_GGG(X, Y, geq_in_gg(X, Y))
U4_GGG(X, Y, notZero_out_g(Y)) → GEQ_IN_GG(X, Y)
GEQ_IN_GG(s(X), s(Y)) → U7_GG(X, Y, geq_in_gg(X, Y))
GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)
REM_IN_GAG(X, Y, X) → U4_GAG(X, Y, notZero_in_a(Y))
REM_IN_GAG(X, Y, X) → NOTZERO_IN_A(Y)
U4_GAG(X, Y, notZero_out_a(Y)) → U5_GAG(X, Y, geq_in_ga(X, Y))
U4_GAG(X, Y, notZero_out_a(Y)) → GEQ_IN_GA(X, Y)
GEQ_IN_GA(s(X), s(Y)) → U7_GA(X, Y, geq_in_ga(X, Y))
GEQ_IN_GA(s(X), s(Y)) → GEQ_IN_GA(X, Y)

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U4_GGG(x1, x2, x3)  =  U4_GGG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x2, x4)
U5_GAG(x1, x2, x3)  =  U5_GAG(x3)
SUB_IN_GAA(x1, x2, x3)  =  SUB_IN_GAA(x1)
GEQ_IN_GG(x1, x2)  =  GEQ_IN_GG(x1, x2)
NOTZERO_IN_A(x1)  =  NOTZERO_IN_A
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
REM_IN_GAG(x1, x2, x3)  =  REM_IN_GAG(x1, x3)
REM_IN_GGG(x1, x2, x3)  =  REM_IN_GGG(x1, x2, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x3, x4)
U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x1, x2, x3, x4)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x4)
NOTZERO_IN_G(x1)  =  NOTZERO_IN_G(x1)
U4_GAG(x1, x2, x3)  =  U4_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x1, x3, x4)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U5_GGG(x1, x2, x3)  =  U5_GGG(x3)

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:

REM_IN_GAG(X, Y, R) → U1_GAG(X, Y, R, notZero_in_a(Y))
REM_IN_GAG(X, Y, R) → NOTZERO_IN_A(Y)
U1_GAG(X, Y, R, notZero_out_a(Y)) → U2_GAG(X, Y, R, sub_in_gaa(X, Y, Z))
U1_GAG(X, Y, R, notZero_out_a(Y)) → SUB_IN_GAA(X, Y, Z)
SUB_IN_GAA(s(X), s(Y), Z) → U6_GAA(X, Y, Z, sub_in_gaa(X, Y, Z))
SUB_IN_GAA(s(X), s(Y), Z) → SUB_IN_GAA(X, Y, Z)
U2_GAG(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_GAG(X, Y, R, rem_in_ggg(Z, Y, R))
U2_GAG(X, Y, R, sub_out_gaa(X, Y, Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y))
REM_IN_GGG(X, Y, R) → NOTZERO_IN_G(Y)
U1_GGG(X, Y, R, notZero_out_g(Y)) → U2_GGG(X, Y, R, sub_in_gga(X, Y, Z))
U1_GGG(X, Y, R, notZero_out_g(Y)) → SUB_IN_GGA(X, Y, Z)
SUB_IN_GGA(s(X), s(Y), Z) → U6_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)
U2_GGG(X, Y, R, sub_out_gga(X, Y, Z)) → U3_GGG(X, Y, R, rem_in_ggg(Z, Y, R))
U2_GGG(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, X) → U4_GGG(X, Y, notZero_in_g(Y))
REM_IN_GGG(X, Y, X) → NOTZERO_IN_G(Y)
U4_GGG(X, Y, notZero_out_g(Y)) → U5_GGG(X, Y, geq_in_gg(X, Y))
U4_GGG(X, Y, notZero_out_g(Y)) → GEQ_IN_GG(X, Y)
GEQ_IN_GG(s(X), s(Y)) → U7_GG(X, Y, geq_in_gg(X, Y))
GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)
REM_IN_GAG(X, Y, X) → U4_GAG(X, Y, notZero_in_a(Y))
REM_IN_GAG(X, Y, X) → NOTZERO_IN_A(Y)
U4_GAG(X, Y, notZero_out_a(Y)) → U5_GAG(X, Y, geq_in_ga(X, Y))
U4_GAG(X, Y, notZero_out_a(Y)) → GEQ_IN_GA(X, Y)
GEQ_IN_GA(s(X), s(Y)) → U7_GA(X, Y, geq_in_ga(X, Y))
GEQ_IN_GA(s(X), s(Y)) → GEQ_IN_GA(X, Y)

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U4_GGG(x1, x2, x3)  =  U4_GGG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x2, x4)
U5_GAG(x1, x2, x3)  =  U5_GAG(x3)
SUB_IN_GAA(x1, x2, x3)  =  SUB_IN_GAA(x1)
GEQ_IN_GG(x1, x2)  =  GEQ_IN_GG(x1, x2)
NOTZERO_IN_A(x1)  =  NOTZERO_IN_A
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
REM_IN_GAG(x1, x2, x3)  =  REM_IN_GAG(x1, x3)
REM_IN_GGG(x1, x2, x3)  =  REM_IN_GGG(x1, x2, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x3, x4)
U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x1, x2, x3, x4)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x4)
NOTZERO_IN_G(x1)  =  NOTZERO_IN_G(x1)
U4_GAG(x1, x2, x3)  =  U4_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x1, x3, x4)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U5_GGG(x1, x2, x3)  =  U5_GGG(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 5 SCCs with 21 less nodes.

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

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

GEQ_IN_GA(s(X), s(Y)) → GEQ_IN_GA(X, Y)

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
GEQ_IN_GA(x1, x2)  =  GEQ_IN_GA(x1)

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

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

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

GEQ_IN_GA(s(X), s(Y)) → GEQ_IN_GA(X, Y)

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

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

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

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

GEQ_IN_GA(s(X)) → GEQ_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



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

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

GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
GEQ_IN_GG(x1, x2)  =  GEQ_IN_GG(x1, x2)

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

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

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

GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)

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

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

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

GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



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

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

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

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

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

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

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

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

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

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

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

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

SUB_IN_GGA(s(X), s(Y)) → SUB_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



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

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

U1_GGG(X, Y, R, notZero_out_g(Y)) → U2_GGG(X, Y, R, sub_in_gga(X, Y, Z))
U2_GGG(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y))

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
REM_IN_GGG(x1, x2, x3)  =  REM_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x1, x2, x3, x4)

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

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

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

U1_GGG(X, Y, R, notZero_out_g(Y)) → U2_GGG(X, Y, R, sub_in_gga(X, Y, Z))
U2_GGG(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y))

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
notZero_in_g(s(X)) → notZero_out_g(s(X))
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
REM_IN_GGG(x1, x2, x3)  =  REM_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x1, x2, x3, x4)

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

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

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

U1_GGG(X, Y, R, notZero_out_g) → U2_GGG(Y, R, sub_in_gga(X, Y))
U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y))

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
notZero_in_g(s(X)) → notZero_out_g
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_GGG(X, Y, R, notZero_out_g) → U2_GGG(Y, R, sub_in_gga(X, Y)) at position [2] we obtained the following new rules:

U1_GGG(x0, 0, y2, notZero_out_g) → U2_GGG(0, y2, sub_out_gga(x0))
U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))



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

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U1_GGG(x0, 0, y2, notZero_out_g) → U2_GGG(0, y2, sub_out_gga(x0))
U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y))

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
notZero_in_g(s(X)) → notZero_out_g
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REM_IN_GGG(X, Y, R) → U1_GGG(X, Y, R, notZero_in_g(Y)) at position [3] we obtained the following new rules:

REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g)



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

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R)
U1_GGG(x0, 0, y2, notZero_out_g) → U2_GGG(0, y2, sub_out_gga(x0))
REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
notZero_in_g(s(X)) → notZero_out_g
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

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

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
notZero_in_g(s(X)) → notZero_out_g
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
              ↳ PiDP

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

notZero_in_g(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Instantiation
              ↳ PiDP

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R)
REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGG(Y, R, sub_out_gga(Z)) → REM_IN_GGG(Z, Y, R) we obtained the following new rules:

U2_GGG(s(z1), z2, sub_out_gga(x2)) → REM_IN_GGG(x2, s(z1), z2)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ ForwardInstantiation
              ↳ PiDP

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(s(z1), z2, sub_out_gga(x2)) → REM_IN_GGG(x2, s(z1), z2)
REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REM_IN_GGG(y0, s(x0), y2) → U1_GGG(y0, s(x0), y2, notZero_out_g) we obtained the following new rules:

REM_IN_GGG(s(y_0), s(x1), x2) → U1_GGG(s(y_0), s(x1), x2, notZero_out_g)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ ForwardInstantiation
                                                  ↳ QDP
                                                    ↳ ForwardInstantiation
              ↳ PiDP

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(s(z1), z2, sub_out_gga(x2)) → REM_IN_GGG(x2, s(z1), z2)
REM_IN_GGG(s(y_0), s(x1), x2) → U1_GGG(s(y_0), s(x1), x2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_GGG(s(z1), z2, sub_out_gga(x2)) → REM_IN_GGG(x2, s(z1), z2) we obtained the following new rules:

U2_GGG(s(x0), x1, sub_out_gga(s(y_0))) → REM_IN_GGG(s(y_0), s(x0), x1)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ ForwardInstantiation
                                                  ↳ QDP
                                                    ↳ ForwardInstantiation
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
              ↳ PiDP

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

U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
U2_GGG(s(x0), x1, sub_out_gga(s(y_0))) → REM_IN_GGG(s(y_0), s(x0), x1)
REM_IN_GGG(s(y_0), s(x1), x2) → U1_GGG(s(y_0), s(x1), x2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U1_GGG(s(x0), s(x1), y2, notZero_out_g) → U2_GGG(s(x1), y2, U6_gga(sub_in_gga(x0, x1)))
The remaining pairs can at least be oriented weakly.

U2_GGG(s(x0), x1, sub_out_gga(s(y_0))) → REM_IN_GGG(s(y_0), s(x0), x1)
REM_IN_GGG(s(y_0), s(x1), x2) → U1_GGG(s(y_0), s(x1), x2, notZero_out_g)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(REM_IN_GGG(x1, x2, x3)) = x1   
POL(U1_GGG(x1, x2, x3, x4)) = x1   
POL(U2_GGG(x1, x2, x3)) = x3   
POL(U6_gga(x1)) = x1   
POL(notZero_out_g) = 0   
POL(s(x1)) = 1 + x1   
POL(sub_in_gga(x1, x2)) = x1   
POL(sub_out_gga(x1)) = x1   

The following usable rules [17] were oriented:

sub_in_gga(X, 0) → sub_out_gga(X)
sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ UsableRulesProof
                                      ↳ QDP
                                        ↳ QReductionProof
                                          ↳ QDP
                                            ↳ Instantiation
                                              ↳ QDP
                                                ↳ ForwardInstantiation
                                                  ↳ QDP
                                                    ↳ ForwardInstantiation
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
              ↳ PiDP

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

U2_GGG(s(x0), x1, sub_out_gga(s(y_0))) → REM_IN_GGG(s(y_0), s(x0), x1)
REM_IN_GGG(s(y_0), s(x1), x2) → U1_GGG(s(y_0), s(x1), x2, notZero_out_g)

The TRS R consists of the following rules:

sub_in_gga(s(X), s(Y)) → U6_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
U6_gga(sub_out_gga(Z)) → sub_out_gga(Z)

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

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

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

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

The TRS R consists of the following rules:

rem_in_gag(X, Y, R) → U1_gag(X, Y, R, notZero_in_a(Y))
notZero_in_a(s(X)) → notZero_out_a(s(X))
U1_gag(X, Y, R, notZero_out_a(Y)) → U2_gag(X, Y, R, sub_in_gaa(X, Y, Z))
sub_in_gaa(s(X), s(Y), Z) → U6_gaa(X, Y, Z, sub_in_gaa(X, Y, Z))
sub_in_gaa(X, 0, X) → sub_out_gaa(X, 0, X)
U6_gaa(X, Y, Z, sub_out_gaa(X, Y, Z)) → sub_out_gaa(s(X), s(Y), Z)
U2_gag(X, Y, R, sub_out_gaa(X, Y, Z)) → U3_gag(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, R) → U1_ggg(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_ggg(X, Y, R, notZero_out_g(Y)) → U2_ggg(X, Y, R, sub_in_gga(X, Y, Z))
sub_in_gga(s(X), s(Y), Z) → U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U2_ggg(X, Y, R, sub_out_gga(X, Y, Z)) → U3_ggg(X, Y, R, rem_in_ggg(Z, Y, R))
rem_in_ggg(X, Y, X) → U4_ggg(X, Y, notZero_in_g(Y))
U4_ggg(X, Y, notZero_out_g(Y)) → U5_ggg(X, Y, geq_in_gg(X, Y))
geq_in_gg(s(X), s(Y)) → U7_gg(X, Y, geq_in_gg(X, Y))
geq_in_gg(X, 0) → geq_out_gg(X, 0)
U7_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U5_ggg(X, Y, geq_out_gg(X, Y)) → rem_out_ggg(X, Y, X)
U3_ggg(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_ggg(X, Y, R)
U3_gag(X, Y, R, rem_out_ggg(Z, Y, R)) → rem_out_gag(X, Y, R)
rem_in_gag(X, Y, X) → U4_gag(X, Y, notZero_in_a(Y))
U4_gag(X, Y, notZero_out_a(Y)) → U5_gag(X, Y, geq_in_ga(X, Y))
geq_in_ga(s(X), s(Y)) → U7_ga(X, Y, geq_in_ga(X, Y))
geq_in_ga(X, 0) → geq_out_ga(X, 0)
U7_ga(X, Y, geq_out_ga(X, Y)) → geq_out_ga(s(X), s(Y))
U5_gag(X, Y, geq_out_ga(X, Y)) → rem_out_gag(X, Y, X)

The argument filtering Pi contains the following mapping:
rem_in_gag(x1, x2, x3)  =  rem_in_gag(x1, x3)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
notZero_in_a(x1)  =  notZero_in_a
notZero_out_a(x1)  =  notZero_out_a
U2_gag(x1, x2, x3, x4)  =  U2_gag(x3, x4)
sub_in_gaa(x1, x2, x3)  =  sub_in_gaa(x1)
s(x1)  =  s(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
sub_out_gaa(x1, x2, x3)  =  sub_out_gaa(x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x2, x4)
rem_in_ggg(x1, x2, x3)  =  rem_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
notZero_out_g(x1)  =  notZero_out_g
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
U4_ggg(x1, x2, x3)  =  U4_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3)  =  U5_ggg(x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
geq_out_gg(x1, x2)  =  geq_out_gg
rem_out_ggg(x1, x2, x3)  =  rem_out_ggg
rem_out_gag(x1, x2, x3)  =  rem_out_gag(x2)
U4_gag(x1, x2, x3)  =  U4_gag(x1, x3)
U5_gag(x1, x2, x3)  =  U5_gag(x3)
geq_in_ga(x1, x2)  =  geq_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
geq_out_ga(x1, x2)  =  geq_out_ga(x2)
SUB_IN_GAA(x1, x2, x3)  =  SUB_IN_GAA(x1)

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

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

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

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

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

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

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

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

SUB_IN_GAA(s(X)) → SUB_IN_GAA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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