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

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇔)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔)
        ↳22 YES
    ↳23 PiDP
        ↳24 UsableRulesProof (⇔)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇐)
        ↳27 QDP
        ↳28 QDPSizeChangeProof (⇔)
        ↳29 YES
    ↳30 PiDP
        ↳31 UsableRulesProof (⇔)
        ↳32 PiDP
        ↳33 PiDPToQDPProof (⇐)
        ↳34 QDP
        ↳35 QDPOrderProof (⇔)
        ↳36 QDP
        ↳37 DependencyGraphProof (⇔)
        ↳38 TRUE
    ↳39 PiDP
        ↳40 UsableRulesProof (⇔)
        ↳41 PiDP
        ↳42 PiDPToQDPProof (⇐)
        ↳43 QDP
        ↳44 QDPSizeChangeProof (⇔)
        ↳45 YES

(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).

Queries:

rem(g,a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

sub14(s(T44), s(T46), X68) :- sub14(T44, T46, X68).
sub14(T51, 0, T51).
geq34(s(T92), s(T94)) :- geq34(T92, T94).
geq34(T99, 0).
sub9(s(T31), T33, X41) :- sub14(T31, T33, X41).
rem1(T7, s(T16), T9) :- sub9(T7, T16, X7).
rem1(T7, s(T20), T9) :- ','(sub9(T7, T20, T19), rem1(T19, s(T20), T9)).
rem1(s(T79), s(T81), s(T79)) :- geq34(T79, T81).

Queries:

rem1(g,a,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rem1_in: (b,f,b) (b,b,b)
sub9_in: (b,f,f) (b,b,f)
sub14_in: (b,f,f) (b,b,f)
geq34_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:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)

(5) 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:

REM1_IN_GAG(T7, s(T16), T9) → U4_GAG(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
REM1_IN_GAG(T7, s(T16), T9) → SUB9_IN_GAA(T7, T16, X7)
SUB9_IN_GAA(s(T31), T33, X41) → U3_GAA(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
SUB9_IN_GAA(s(T31), T33, X41) → SUB14_IN_GAA(T31, T33, X41)
SUB14_IN_GAA(s(T44), s(T46), X68) → U1_GAA(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
SUB14_IN_GAA(s(T44), s(T46), X68) → SUB14_IN_GAA(T44, T46, X68)
REM1_IN_GAG(T7, s(T20), T9) → U5_GAG(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_GAG(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_GAG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U5_GAG(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(T7, s(T16), T9) → U4_GGG(T7, T16, T9, sub9_in_gga(T7, T16, X7))
REM1_IN_GGG(T7, s(T16), T9) → SUB9_IN_GGA(T7, T16, X7)
SUB9_IN_GGA(s(T31), T33, X41) → U3_GGA(T31, T33, X41, sub14_in_gga(T31, T33, X41))
SUB9_IN_GGA(s(T31), T33, X41) → SUB14_IN_GGA(T31, T33, X41)
SUB14_IN_GGA(s(T44), s(T46), X68) → U1_GGA(T44, T46, X68, sub14_in_gga(T44, T46, X68))
SUB14_IN_GGA(s(T44), s(T46), X68) → SUB14_IN_GGA(T44, T46, X68)
REM1_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_GGG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U5_GGG(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(s(T79), s(T81), s(T79)) → U7_GGG(T79, T81, geq34_in_gg(T79, T81))
REM1_IN_GGG(s(T79), s(T81), s(T79)) → GEQ34_IN_GG(T79, T81)
GEQ34_IN_GG(s(T92), s(T94)) → U2_GG(T92, T94, geq34_in_gg(T92, T94))
GEQ34_IN_GG(s(T92), s(T94)) → GEQ34_IN_GG(T92, T94)
REM1_IN_GAG(s(T79), s(T81), s(T79)) → U7_GAG(T79, T81, geq34_in_ga(T79, T81))
REM1_IN_GAG(s(T79), s(T81), s(T79)) → GEQ34_IN_GA(T79, T81)
GEQ34_IN_GA(s(T92), s(T94)) → U2_GA(T92, T94, geq34_in_ga(T92, T94))
GEQ34_IN_GA(s(T92), s(T94)) → GEQ34_IN_GA(T92, T94)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
REM1_IN_GAG(x1, x2, x3)  =  REM1_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x4)
SUB9_IN_GAA(x1, x2, x3)  =  SUB9_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(x3, x4)
U6_GAG(x1, x2, x3, x4)  =  U6_GAG(x2, x4)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x4)
SUB9_IN_GGA(x1, x2, x3)  =  SUB9_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x2, x3, x4)
U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x4)
U7_GGG(x1, x2, x3)  =  U7_GGG(x3)
GEQ34_IN_GG(x1, x2)  =  GEQ34_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U7_GAG(x1, x2, x3)  =  U7_GAG(x3)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(6) Obligation:

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

REM1_IN_GAG(T7, s(T16), T9) → U4_GAG(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
REM1_IN_GAG(T7, s(T16), T9) → SUB9_IN_GAA(T7, T16, X7)
SUB9_IN_GAA(s(T31), T33, X41) → U3_GAA(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
SUB9_IN_GAA(s(T31), T33, X41) → SUB14_IN_GAA(T31, T33, X41)
SUB14_IN_GAA(s(T44), s(T46), X68) → U1_GAA(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
SUB14_IN_GAA(s(T44), s(T46), X68) → SUB14_IN_GAA(T44, T46, X68)
REM1_IN_GAG(T7, s(T20), T9) → U5_GAG(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_GAG(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_GAG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U5_GAG(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(T7, s(T16), T9) → U4_GGG(T7, T16, T9, sub9_in_gga(T7, T16, X7))
REM1_IN_GGG(T7, s(T16), T9) → SUB9_IN_GGA(T7, T16, X7)
SUB9_IN_GGA(s(T31), T33, X41) → U3_GGA(T31, T33, X41, sub14_in_gga(T31, T33, X41))
SUB9_IN_GGA(s(T31), T33, X41) → SUB14_IN_GGA(T31, T33, X41)
SUB14_IN_GGA(s(T44), s(T46), X68) → U1_GGA(T44, T46, X68, sub14_in_gga(T44, T46, X68))
SUB14_IN_GGA(s(T44), s(T46), X68) → SUB14_IN_GGA(T44, T46, X68)
REM1_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_GGG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U5_GGG(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(s(T79), s(T81), s(T79)) → U7_GGG(T79, T81, geq34_in_gg(T79, T81))
REM1_IN_GGG(s(T79), s(T81), s(T79)) → GEQ34_IN_GG(T79, T81)
GEQ34_IN_GG(s(T92), s(T94)) → U2_GG(T92, T94, geq34_in_gg(T92, T94))
GEQ34_IN_GG(s(T92), s(T94)) → GEQ34_IN_GG(T92, T94)
REM1_IN_GAG(s(T79), s(T81), s(T79)) → U7_GAG(T79, T81, geq34_in_ga(T79, T81))
REM1_IN_GAG(s(T79), s(T81), s(T79)) → GEQ34_IN_GA(T79, T81)
GEQ34_IN_GA(s(T92), s(T94)) → U2_GA(T92, T94, geq34_in_ga(T92, T94))
GEQ34_IN_GA(s(T92), s(T94)) → GEQ34_IN_GA(T92, T94)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
REM1_IN_GAG(x1, x2, x3)  =  REM1_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x4)
SUB9_IN_GAA(x1, x2, x3)  =  SUB9_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(x3, x4)
U6_GAG(x1, x2, x3, x4)  =  U6_GAG(x2, x4)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x4)
SUB9_IN_GGA(x1, x2, x3)  =  SUB9_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x2, x3, x4)
U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x4)
U7_GGG(x1, x2, x3)  =  U7_GGG(x3)
GEQ34_IN_GG(x1, x2)  =  GEQ34_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U7_GAG(x1, x2, x3)  =  U7_GAG(x3)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 20 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

GEQ34_IN_GA(s(T92), s(T94)) → GEQ34_IN_GA(T92, T94)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GEQ34_IN_GA(s(T92), s(T94)) → GEQ34_IN_GA(T92, T94)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

GEQ34_IN_GA(s(T92)) → GEQ34_IN_GA(T92)

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

(14) 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:

  • GEQ34_IN_GA(s(T92)) → GEQ34_IN_GA(T92)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

GEQ34_IN_GG(s(T92), s(T94)) → GEQ34_IN_GG(T92, T94)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
GEQ34_IN_GG(x1, x2)  =  GEQ34_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

GEQ34_IN_GG(s(T92), s(T94)) → GEQ34_IN_GG(T92, T94)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

GEQ34_IN_GG(s(T92), s(T94)) → GEQ34_IN_GG(T92, T94)

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

(21) 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:

  • GEQ34_IN_GG(s(T92), s(T94)) → GEQ34_IN_GG(T92, T94)
    The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

SUB14_IN_GGA(s(T44), s(T46), X68) → SUB14_IN_GGA(T44, T46, X68)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

SUB14_IN_GGA(s(T44), s(T46), X68) → SUB14_IN_GGA(T44, T46, X68)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

SUB14_IN_GGA(s(T44), s(T46)) → SUB14_IN_GGA(T44, T46)

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

(28) 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:

  • SUB14_IN_GGA(s(T44), s(T46)) → SUB14_IN_GGA(T44, T46)
    The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x2, x3, x4)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x2, x3, x4)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U5_GGG(T20, T9, sub9_in_gga(T7, T20))
U5_GGG(T20, T9, sub9_out_gga(T19)) → REM1_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

sub9_in_gga(s(T31), T33) → U3_gga(sub14_in_gga(T31, T33))
U3_gga(sub14_out_gga(X41)) → sub9_out_gga(X41)
sub14_in_gga(s(T44), s(T46)) → U1_gga(sub14_in_gga(T44, T46))
sub14_in_gga(T51, 0) → sub14_out_gga(T51)
U1_gga(sub14_out_gga(X68)) → sub14_out_gga(X68)

The set Q consists of the following terms:

sub9_in_gga(x0, x1)
U3_gga(x0)
sub14_in_gga(x0, x1)
U1_gga(x0)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U5_GGG(T20, T9, sub9_out_gga(T19)) → REM1_IN_GGG(T19, s(T20), T9)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(REM1_IN_GGG(x1, x2, x3)) = 1 + x1   
POL(U1_gga(x1)) = x1   
POL(U3_gga(x1)) = x1   
POL(U5_GGG(x1, x2, x3)) = 1 + x3   
POL(s(x1)) = 1 + x1   
POL(sub14_in_gga(x1, x2)) = 1 + x1   
POL(sub14_out_gga(x1)) = 1 + x1   
POL(sub9_in_gga(x1, x2)) = x1   
POL(sub9_out_gga(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

sub9_in_gga(s(T31), T33) → U3_gga(sub14_in_gga(T31, T33))
sub14_in_gga(s(T44), s(T46)) → U1_gga(sub14_in_gga(T44, T46))
sub14_in_gga(T51, 0) → sub14_out_gga(T51)
U3_gga(sub14_out_gga(X41)) → sub9_out_gga(X41)
U1_gga(sub14_out_gga(X68)) → sub14_out_gga(X68)

(36) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U5_GGG(T20, T9, sub9_in_gga(T7, T20))

The TRS R consists of the following rules:

sub9_in_gga(s(T31), T33) → U3_gga(sub14_in_gga(T31, T33))
U3_gga(sub14_out_gga(X41)) → sub9_out_gga(X41)
sub14_in_gga(s(T44), s(T46)) → U1_gga(sub14_in_gga(T44, T46))
sub14_in_gga(T51, 0) → sub14_out_gga(T51)
U1_gga(sub14_out_gga(X68)) → sub14_out_gga(X68)

The set Q consists of the following terms:

sub9_in_gga(x0, x1)
U3_gga(x0)
sub14_in_gga(x0, x1)
U1_gga(x0)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(38) TRUE

(39) Obligation:

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

SUB14_IN_GAA(s(T44), s(T46), X68) → SUB14_IN_GAA(T44, T46, X68)

The TRS R consists of the following rules:

rem1_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, sub9_in_gaa(T7, T16, X7))
sub9_in_gaa(s(T31), T33, X41) → U3_gaa(T31, T33, X41, sub14_in_gaa(T31, T33, X41))
sub14_in_gaa(s(T44), s(T46), X68) → U1_gaa(T44, T46, X68, sub14_in_gaa(T44, T46, X68))
sub14_in_gaa(T51, 0, T51) → sub14_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X68, sub14_out_gaa(T44, T46, X68)) → sub14_out_gaa(s(T44), s(T46), X68)
U3_gaa(T31, T33, X41, sub14_out_gaa(T31, T33, X41)) → sub9_out_gaa(s(T31), T33, X41)
U4_gag(T7, T16, T9, sub9_out_gaa(T7, T16, X7)) → rem1_out_gag(T7, s(T16), T9)
rem1_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, sub9_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, sub9_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, sub9_in_gga(T7, T16, X7))
sub9_in_gga(s(T31), T33, X41) → U3_gga(T31, T33, X41, sub14_in_gga(T31, T33, X41))
sub14_in_gga(s(T44), s(T46), X68) → U1_gga(T44, T46, X68, sub14_in_gga(T44, T46, X68))
sub14_in_gga(T51, 0, T51) → sub14_out_gga(T51, 0, T51)
U1_gga(T44, T46, X68, sub14_out_gga(T44, T46, X68)) → sub14_out_gga(s(T44), s(T46), X68)
U3_gga(T31, T33, X41, sub14_out_gga(T31, T33, X41)) → sub9_out_gga(s(T31), T33, X41)
U4_ggg(T7, T16, T9, sub9_out_gga(T7, T16, X7)) → rem1_out_ggg(T7, s(T16), T9)
rem1_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, sub9_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, sub9_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
rem1_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geq34_in_gg(T79, T81))
geq34_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geq34_in_gg(T92, T94))
geq34_in_gg(T99, 0) → geq34_out_gg(T99, 0)
U2_gg(T92, T94, geq34_out_gg(T92, T94)) → geq34_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geq34_out_gg(T79, T81)) → rem1_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, rem1_out_ggg(T19, s(T20), T9)) → rem1_out_gag(T7, s(T20), T9)
rem1_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geq34_in_ga(T79, T81))
geq34_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geq34_in_ga(T92, T94))
geq34_in_ga(T99, 0) → geq34_out_ga(T99, 0)
U2_ga(T92, T94, geq34_out_ga(T92, T94)) → geq34_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geq34_out_ga(T79, T81)) → rem1_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
rem1_in_gag(x1, x2, x3)  =  rem1_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
sub9_in_gaa(x1, x2, x3)  =  sub9_in_gaa(x1)
s(x1)  =  s(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
sub14_out_gaa(x1, x2, x3)  =  sub14_out_gaa(x2, x3)
sub9_out_gaa(x1, x2, x3)  =  sub9_out_gaa(x2, x3)
rem1_out_gag(x1, x2, x3)  =  rem1_out_gag(x2)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4)  =  U6_gag(x2, x4)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4)  =  U4_ggg(x4)
sub9_in_gga(x1, x2, x3)  =  sub9_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
sub14_out_gga(x1, x2, x3)  =  sub14_out_gga(x3)
sub9_out_gga(x1, x2, x3)  =  sub9_out_gga(x3)
rem1_out_ggg(x1, x2, x3)  =  rem1_out_ggg
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x4)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq34_out_gg(x1, x2)  =  geq34_out_gg
U7_gag(x1, x2, x3)  =  U7_gag(x3)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
geq34_out_ga(x1, x2)  =  geq34_out_ga(x2)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)

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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

SUB14_IN_GAA(s(T44), s(T46), X68) → SUB14_IN_GAA(T44, T46, X68)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

SUB14_IN_GAA(s(T44)) → SUB14_IN_GAA(T44)

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

(44) 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:

  • SUB14_IN_GAA(s(T44)) → SUB14_IN_GAA(T44)
    The graph contains the following edges 1 > 1

(45) YES