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

0 Prolog
↳1 PrologToPiTRSProof (⇒, 0 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 13 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 Narrowing (⇒, 0 ms)
        ↳27 QDP
        ↳28 UsableRulesProof (⇔, 0 ms)
        ↳29 QDP
        ↳30 QReductionProof (⇔, 5 ms)
        ↳31 QDP
        ↳32 QDPOrderProof (⇔, 26 ms)
        ↳33 QDP
        ↳34 DependencyGraphProof (⇔, 0 ms)
        ↳35 TRUE

(0) Obligation:

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).

Query: rem(g,g,a)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
rem_in: (b,b,f)
sub_in: (b,b,f)
geq_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rem_in_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

rem_in_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

REM_IN_GGA(X, Y, R) → U1_GGA(X, Y, R, notZero_in_g(Y))
REM_IN_GGA(X, Y, R) → NOTZERO_IN_G(Y)
U1_GGA(X, Y, R, notZero_out_g(Y)) → U2_GGA(X, Y, R, sub_in_gga(X, Y, Z))
U1_GGA(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_GGA(X, Y, R, sub_out_gga(X, Y, Z)) → U3_GGA(X, Y, R, rem_in_gga(Z, Y, R))
U2_GGA(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y, R)
REM_IN_GGA(X, Y, X) → U4_GGA(X, Y, notZero_in_g(Y))
REM_IN_GGA(X, Y, X) → NOTZERO_IN_G(Y)
U4_GGA(X, Y, notZero_out_g(Y)) → U5_GGA(X, Y, geq_in_gg(X, Y))
U4_GGA(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)

The TRS R consists of the following rules:

rem_in_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)
REM_IN_GGA(x1, x2, x3)  =  REM_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
NOTZERO_IN_G(x1)  =  NOTZERO_IN_G(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3)  =  U4_GGA(x1, x2, x3)
U5_GGA(x1, x2, x3)  =  U5_GGA(x1, x2, x3)
GEQ_IN_GG(x1, x2)  =  GEQ_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)

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

(4) Obligation:

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

REM_IN_GGA(X, Y, R) → U1_GGA(X, Y, R, notZero_in_g(Y))
REM_IN_GGA(X, Y, R) → NOTZERO_IN_G(Y)
U1_GGA(X, Y, R, notZero_out_g(Y)) → U2_GGA(X, Y, R, sub_in_gga(X, Y, Z))
U1_GGA(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_GGA(X, Y, R, sub_out_gga(X, Y, Z)) → U3_GGA(X, Y, R, rem_in_gga(Z, Y, R))
U2_GGA(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y, R)
REM_IN_GGA(X, Y, X) → U4_GGA(X, Y, notZero_in_g(Y))
REM_IN_GGA(X, Y, X) → NOTZERO_IN_G(Y)
U4_GGA(X, Y, notZero_out_g(Y)) → U5_GGA(X, Y, geq_in_gg(X, Y))
U4_GGA(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)

The TRS R consists of the following rules:

rem_in_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)
REM_IN_GGA(x1, x2, x3)  =  REM_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
NOTZERO_IN_G(x1)  =  NOTZERO_IN_G(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3)  =  U4_GGA(x1, x2, x3)
U5_GGA(x1, x2, x3)  =  U5_GGA(x1, x2, x3)
GEQ_IN_GG(x1, x2)  =  GEQ_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)
GEQ_IN_GG(x1, x2)  =  GEQ_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(13) YES

(14) Obligation:

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_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

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.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(20) YES

(21) Obligation:

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

U1_GGA(X, Y, R, notZero_out_g(Y)) → U2_GGA(X, Y, R, sub_in_gga(X, Y, Z))
U2_GGA(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y, R)
REM_IN_GGA(X, Y, R) → U1_GGA(X, Y, R, notZero_in_g(Y))

The TRS R consists of the following rules:

rem_in_gga(X, Y, R) → U1_gga(X, Y, R, notZero_in_g(Y))
notZero_in_g(s(X)) → notZero_out_g(s(X))
U1_gga(X, Y, R, notZero_out_g(Y)) → U2_gga(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_gga(X, Y, R, sub_out_gga(X, Y, Z)) → U3_gga(X, Y, R, rem_in_gga(Z, Y, R))
rem_in_gga(X, Y, X) → U4_gga(X, Y, notZero_in_g(Y))
U4_gga(X, Y, notZero_out_g(Y)) → U5_gga(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_gga(X, Y, geq_out_gg(X, Y)) → rem_out_gga(X, Y, X)
U3_gga(X, Y, R, rem_out_gga(Z, Y, R)) → rem_out_gga(X, Y, R)

The argument filtering Pi contains the following mapping:
rem_in_gga(x1, x2, x3)  =  rem_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3)  =  U4_gga(x1, x2, x3)
U5_gga(x1, x2, x3)  =  U5_gga(x1, x2, x3)
geq_in_gg(x1, x2)  =  geq_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
geq_out_gg(x1, x2)  =  geq_out_gg(x1, x2)
rem_out_gga(x1, x2, x3)  =  rem_out_gga(x1, x2, x3)
REM_IN_GGA(x1, x2, x3)  =  REM_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U1_GGA(X, Y, R, notZero_out_g(Y)) → U2_GGA(X, Y, R, sub_in_gga(X, Y, Z))
U2_GGA(X, Y, R, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y, R)
REM_IN_GGA(X, Y, R) → U1_GGA(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:
notZero_in_g(x1)  =  notZero_in_g(x1)
s(x1)  =  s(x1)
notZero_out_g(x1)  =  notZero_out_g(x1)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
0  =  0
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
REM_IN_GGA(x1, x2, x3)  =  REM_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U1_GGA(X, Y, notZero_out_g(Y)) → U2_GGA(X, Y, sub_in_gga(X, Y))
U2_GGA(X, Y, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y)
REM_IN_GGA(X, Y) → U1_GGA(X, Y, notZero_in_g(Y))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0, x1, x2)

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

(26) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REM_IN_GGA(X, Y) → U1_GGA(X, Y, notZero_in_g(Y)) at position [2] we obtained the following new rules [LPAR04]:

REM_IN_GGA(y0, s(x0)) → U1_GGA(y0, s(x0), notZero_out_g(s(x0)))

(27) Obligation:

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

U1_GGA(X, Y, notZero_out_g(Y)) → U2_GGA(X, Y, sub_in_gga(X, Y))
U2_GGA(X, Y, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y)
REM_IN_GGA(y0, s(x0)) → U1_GGA(y0, s(x0), notZero_out_g(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0, x1, x2)

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

(28) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(29) Obligation:

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

U1_GGA(X, Y, notZero_out_g(Y)) → U2_GGA(X, Y, sub_in_gga(X, Y))
U2_GGA(X, Y, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y)
REM_IN_GGA(y0, s(x0)) → U1_GGA(y0, s(x0), notZero_out_g(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

sub_in_gga(x0, x1)
notZero_in_g(x0)
U6_gga(x0, x1, x2)

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

(30) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

notZero_in_g(x0)

(31) Obligation:

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

U1_GGA(X, Y, notZero_out_g(Y)) → U2_GGA(X, Y, sub_in_gga(X, Y))
U2_GGA(X, Y, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y)
REM_IN_GGA(y0, s(x0)) → U1_GGA(y0, s(x0), notZero_out_g(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0, x1, x2)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


REM_IN_GGA(y0, s(x0)) → U1_GGA(y0, s(x0), notZero_out_g(s(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(REM_IN_GGA(x1, x2)) = 1 + x1 + x2   
POL(U1_GGA(x1, x2, x3)) = 1 + x1   
POL(U2_GGA(x1, x2, x3)) = x3   
POL(U6_gga(x1, x2, x3)) = 1 + x3   
POL(notZero_out_g(x1)) = 0   
POL(s(x1)) = 1 + x1   
POL(sub_in_gga(x1, x2)) = 1 + x1   
POL(sub_out_gga(x1, x2, x3)) = 1 + x2 + x3   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(33) Obligation:

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

U1_GGA(X, Y, notZero_out_g(Y)) → U2_GGA(X, Y, sub_in_gga(X, Y))
U2_GGA(X, Y, sub_out_gga(X, Y, Z)) → REM_IN_GGA(Z, Y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

sub_in_gga(x0, x1)
U6_gga(x0, x1, x2)

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

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(35) TRUE