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

Built DT problem from termination graph.

(2) Obligation:

Triples:

sub14(s(T44), s(T46), X68) :- sub14(T44, T46, X68).
geq34(s(T92), s(T94)) :- geq34(T92, T94).
rem1(s(T31), s(T33), T9) :- sub14(T31, T33, X41).
rem1(T7, s(T20), T9) :- ','(subc9(T7, T20, T19), rem1(T19, s(T20), T9)).
rem1(s(T79), s(T81), s(T79)) :- geq34(T79, T81).

Clauses:

remc1(T7, s(T20), T9) :- ','(subc9(T7, T20, T19), remc1(T19, s(T20), T9)).
remc1(s(T79), s(T81), s(T79)) :- geqc34(T79, T81).
subc14(s(T44), s(T46), X68) :- subc14(T44, T46, X68).
subc14(T51, 0, T51).
geqc34(s(T92), s(T94)) :- geqc34(T92, T94).
geqc34(T99, 0).
subc9(s(T31), T33, X41) :- subc14(T31, T33, X41).

Afs:

rem1(x1, x2, x3)  =  rem1(x1, x3)

(3) TriplesToPiDPProof (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)
sub14_in: (b,f,f) (b,b,f)
subc9_in: (b,f,f) (b,b,f)
subc14_in: (b,f,f) (b,b,f)
geq34_in: (b,b) (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REM1_IN_GAG(s(T31), s(T33), T9) → U3_GAG(T31, T33, T9, sub14_in_gaa(T31, T33, X41))
REM1_IN_GAG(s(T31), s(T33), T9) → 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) → U4_GAG(T7, T20, T9, subc9_in_gaa(T7, T20, T19))
U4_GAG(T7, T20, T9, subc9_out_gaa(T7, T20, T19)) → U5_GAG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U4_GAG(T7, T20, T9, subc9_out_gaa(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(s(T31), s(T33), T9) → U3_GGG(T31, T33, T9, sub14_in_gga(T31, T33, X41))
REM1_IN_GGG(s(T31), s(T33), T9) → 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) → U4_GGG(T7, T20, T9, subc9_in_gga(T7, T20, T19))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → U5_GGG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(s(T79), s(T81), s(T79)) → U6_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)) → U6_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:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
REM1_IN_GAG(x1, x2, x3)  =  REM1_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x1, x3, x4)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x1, x4)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x1, x3, x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(x1, x2, x3, x4)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x1, x2, x3, x4)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x1, x2, x3, x4)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x1, x2, x3, x4)
U6_GGG(x1, x2, x3)  =  U6_GGG(x1, x2, x3)
GEQ34_IN_GG(x1, x2)  =  GEQ34_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U6_GAG(x1, x2, x3)  =  U6_GAG(x1, x3)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

REM1_IN_GAG(s(T31), s(T33), T9) → U3_GAG(T31, T33, T9, sub14_in_gaa(T31, T33, X41))
REM1_IN_GAG(s(T31), s(T33), T9) → 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) → U4_GAG(T7, T20, T9, subc9_in_gaa(T7, T20, T19))
U4_GAG(T7, T20, T9, subc9_out_gaa(T7, T20, T19)) → U5_GAG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U4_GAG(T7, T20, T9, subc9_out_gaa(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(s(T31), s(T33), T9) → U3_GGG(T31, T33, T9, sub14_in_gga(T31, T33, X41))
REM1_IN_GGG(s(T31), s(T33), T9) → 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) → U4_GGG(T7, T20, T9, subc9_in_gga(T7, T20, T19))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → U5_GGG(T7, T20, T9, rem1_in_ggg(T19, s(T20), T9))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)
REM1_IN_GGG(s(T79), s(T81), s(T79)) → U6_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)) → U6_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:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
sub14_in_gaa(x1, x2, x3)  =  sub14_in_gaa(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
rem1_in_ggg(x1, x2, x3)  =  rem1_in_ggg(x1, x2, x3)
sub14_in_gga(x1, x2, x3)  =  sub14_in_gga(x1, x2)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
geq34_in_gg(x1, x2)  =  geq34_in_gg(x1, x2)
geq34_in_ga(x1, x2)  =  geq34_in_ga(x1)
REM1_IN_GAG(x1, x2, x3)  =  REM1_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x1, x3, x4)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x1, x4)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x1, x3, x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(x1, x2, x3, x4)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x1, x2, x3, x4)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x1, x2, x3, x4)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x1, x2, x3, x4)
U6_GGG(x1, x2, x3)  =  U6_GGG(x1, x2, x3)
GEQ34_IN_GG(x1, x2)  =  GEQ34_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U6_GAG(x1, x2, x3)  =  U6_GAG(x1, x3)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 less nodes.

(6) Complex Obligation (AND)

(7) 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:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
GEQ34_IN_GA(x1, x2)  =  GEQ34_IN_GA(x1)

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:

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

(10) PiDPToQDPProof (SOUND 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:

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

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:

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

(13) YES

(14) 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:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
GEQ34_IN_GG(x1, x2)  =  GEQ34_IN_GG(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:

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

(17) PiDPToQDPProof (EQUIVALENT 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:

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.

(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:

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

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

The TRS R consists of the following rules:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
SUB14_IN_GGA(x1, x2, x3)  =  SUB14_IN_GGA(x1, x2)

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:

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

(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:

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.

(26) 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

(27) YES

(28) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U4_GGG(T7, T20, T9, subc9_in_gga(T7, T20, T19))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x1, x2, x3, x4)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U4_GGG(T7, T20, T9, subc9_in_gga(T7, T20, T19))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
REM1_IN_GGG(x1, x2, x3)  =  REM1_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x1, x2, x3, x4)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U4_GGG(T7, T20, T9, subc9_in_gga(T7, T20))
U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, T19)) → REM1_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

subc9_in_gga(s(T31), T33) → U13_gga(T31, T33, subc14_in_gga(T31, T33))
U13_gga(T31, T33, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)
subc14_in_gga(s(T44), s(T46)) → U11_gga(T44, T46, subc14_in_gga(T44, T46))
subc14_in_gga(T51, 0) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)

The set Q consists of the following terms:

subc9_in_gga(x0, x1)
U13_gga(x0, x1, x2)
subc14_in_gga(x0, x1)
U11_gga(x0, x1, x2)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U4_GGG(T7, T20, T9, subc9_out_gga(T7, T20, 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)) = x1   
POL(U11_gga(x1, x2, x3)) = 1 + x3   
POL(U13_gga(x1, x2, x3)) = x3   
POL(U4_GGG(x1, x2, x3, x4)) = x4   
POL(s(x1)) = 1 + x1   
POL(subc14_in_gga(x1, x2)) = 1 + x1   
POL(subc14_out_gga(x1, x2, x3)) = 1 + x3   
POL(subc9_in_gga(x1, x2)) = x1   
POL(subc9_out_gga(x1, x2, x3)) = 1 + x3   

The following usable rules [FROCOS05] were oriented:

subc9_in_gga(s(T31), T33) → U13_gga(T31, T33, subc14_in_gga(T31, T33))
subc14_in_gga(s(T44), s(T46)) → U11_gga(T44, T46, subc14_in_gga(T44, T46))
subc14_in_gga(T51, 0) → subc14_out_gga(T51, 0, T51)
U13_gga(T31, T33, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)
U11_gga(T44, T46, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)

(34) Obligation:

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

REM1_IN_GGG(T7, s(T20), T9) → U4_GGG(T7, T20, T9, subc9_in_gga(T7, T20))

The TRS R consists of the following rules:

subc9_in_gga(s(T31), T33) → U13_gga(T31, T33, subc14_in_gga(T31, T33))
U13_gga(T31, T33, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)
subc14_in_gga(s(T44), s(T46)) → U11_gga(T44, T46, subc14_in_gga(T44, T46))
subc14_in_gga(T51, 0) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)

The set Q consists of the following terms:

subc9_in_gga(x0, x1)
U13_gga(x0, x1, x2)
subc14_in_gga(x0, x1)
U11_gga(x0, x1, x2)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

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

(36) TRUE

(37) 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:

subc9_in_gaa(s(T31), T33, X41) → U13_gaa(T31, T33, X41, subc14_in_gaa(T31, T33, X41))
subc14_in_gaa(s(T44), s(T46), X68) → U11_gaa(T44, T46, X68, subc14_in_gaa(T44, T46, X68))
subc14_in_gaa(T51, 0, T51) → subc14_out_gaa(T51, 0, T51)
U11_gaa(T44, T46, X68, subc14_out_gaa(T44, T46, X68)) → subc14_out_gaa(s(T44), s(T46), X68)
U13_gaa(T31, T33, X41, subc14_out_gaa(T31, T33, X41)) → subc9_out_gaa(s(T31), T33, X41)
subc9_in_gga(s(T31), T33, X41) → U13_gga(T31, T33, X41, subc14_in_gga(T31, T33, X41))
subc14_in_gga(s(T44), s(T46), X68) → U11_gga(T44, T46, X68, subc14_in_gga(T44, T46, X68))
subc14_in_gga(T51, 0, T51) → subc14_out_gga(T51, 0, T51)
U11_gga(T44, T46, X68, subc14_out_gga(T44, T46, X68)) → subc14_out_gga(s(T44), s(T46), X68)
U13_gga(T31, T33, X41, subc14_out_gga(T31, T33, X41)) → subc9_out_gga(s(T31), T33, X41)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subc9_in_gaa(x1, x2, x3)  =  subc9_in_gaa(x1)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
subc14_in_gaa(x1, x2, x3)  =  subc14_in_gaa(x1)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
subc14_out_gaa(x1, x2, x3)  =  subc14_out_gaa(x1, x2, x3)
subc9_out_gaa(x1, x2, x3)  =  subc9_out_gaa(x1, x2, x3)
subc9_in_gga(x1, x2, x3)  =  subc9_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
subc14_in_gga(x1, x2, x3)  =  subc14_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
0  =  0
subc14_out_gga(x1, x2, x3)  =  subc14_out_gga(x1, x2, x3)
subc9_out_gga(x1, x2, x3)  =  subc9_out_gga(x1, x2, x3)
SUB14_IN_GAA(x1, x2, x3)  =  SUB14_IN_GAA(x1)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(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)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) 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.

(42) 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

(43) YES