(0) Obligation:
Clauses:app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app([], Ys, Ys).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
Queries:
app3_b(g,g,g,a).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:app7([], T25, T25).
app7(.(T34, T35), T36, .(T34, T38)) :- app7(T35, T36, T38).
app19([], T59, T59).
app19(.(T66, T67), T68, .(T66, X85)) :- app19(T67, T68, X85).
app3_b1(T9, [], T18, T13) :- app7(T9, T18, T13).
app3_b1(T9, .(T47, T48), T49, T13) :- app19(T48, T49, X56).
app3_b1(T9, .(T47, T48), T49, T13) :- ','(app19(T48, T49, T52), app7(T9, .(T47, T52), T13)).
Queries:
app3_b1(g,g,g,a).
(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:app3_b1_in: (b,b,b,f)
app7_in: (b,b,f)
app19_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
app3_b1_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, app7_in_gga(T9, T18, T13))
app7_in_gga([], T25, T25) → app7_out_gga([], T25, T25)
app7_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, app7_out_gga(T35, T36, T38)) → app7_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, app7_out_gga(T9, T18, T13)) → app3_b1_out_ggga(T9, [], T18, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
app19_in_gga([], T59, T59) → app19_out_gga([], T59, T59)
app19_in_gga(.(T66, T67), T68, .(T66, X85)) → U2_gga(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
U2_gga(T66, T67, T68, X85, app19_out_gga(T67, T68, X85)) → app19_out_gga(.(T66, T67), T68, .(T66, X85))
U4_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, X56)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, app7_out_gga(T9, .(T47, T52), T13)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
The argument filtering Pi contains the following mapping:
app3_b1_in_ggga(x1, x2, x3, x4) = app3_b1_in_ggga(x1, x2, x3)
[] = []
U3_ggga(x1, x2, x3, x4) = U3_ggga(x4)
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
app7_out_gga(x1, x2, x3) = app7_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app3_b1_out_ggga(x1, x2, x3, x4) = app3_b1_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
app19_out_gga(x1, x2, x3) = app19_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6) = U6_ggga(x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
app3_b1_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, app7_in_gga(T9, T18, T13))
app7_in_gga([], T25, T25) → app7_out_gga([], T25, T25)
app7_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, app7_out_gga(T35, T36, T38)) → app7_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, app7_out_gga(T9, T18, T13)) → app3_b1_out_ggga(T9, [], T18, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
app19_in_gga([], T59, T59) → app19_out_gga([], T59, T59)
app19_in_gga(.(T66, T67), T68, .(T66, X85)) → U2_gga(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
U2_gga(T66, T67, T68, X85, app19_out_gga(T67, T68, X85)) → app19_out_gga(.(T66, T67), T68, .(T66, X85))
U4_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, X56)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, app7_out_gga(T9, .(T47, T52), T13)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
The argument filtering Pi contains the following mapping:
app3_b1_in_ggga(x1, x2, x3, x4) = app3_b1_in_ggga(x1, x2, x3)
[] = []
U3_ggga(x1, x2, x3, x4) = U3_ggga(x4)
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
app7_out_gga(x1, x2, x3) = app7_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app3_b1_out_ggga(x1, x2, x3, x4) = app3_b1_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
app19_out_gga(x1, x2, x3) = app19_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6) = U6_ggga(x6)
(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:
APP3_B1_IN_GGGA(T9, [], T18, T13) → U3_GGGA(T9, T18, T13, app7_in_gga(T9, T18, T13))
APP3_B1_IN_GGGA(T9, [], T18, T13) → APP7_IN_GGA(T9, T18, T13)
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U4_GGGA(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → APP19_IN_GGA(T48, T49, X56)
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → U2_GGA(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U5_GGGA(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_GGGA(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_GGGA(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U5_GGGA(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → APP7_IN_GGA(T9, .(T47, T52), T13)
The TRS R consists of the following rules:
app3_b1_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, app7_in_gga(T9, T18, T13))
app7_in_gga([], T25, T25) → app7_out_gga([], T25, T25)
app7_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, app7_out_gga(T35, T36, T38)) → app7_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, app7_out_gga(T9, T18, T13)) → app3_b1_out_ggga(T9, [], T18, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
app19_in_gga([], T59, T59) → app19_out_gga([], T59, T59)
app19_in_gga(.(T66, T67), T68, .(T66, X85)) → U2_gga(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
U2_gga(T66, T67, T68, X85, app19_out_gga(T67, T68, X85)) → app19_out_gga(.(T66, T67), T68, .(T66, X85))
U4_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, X56)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, app7_out_gga(T9, .(T47, T52), T13)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
The argument filtering Pi contains the following mapping:
app3_b1_in_ggga(x1, x2, x3, x4) = app3_b1_in_ggga(x1, x2, x3)
[] = []
U3_ggga(x1, x2, x3, x4) = U3_ggga(x4)
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
app7_out_gga(x1, x2, x3) = app7_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app3_b1_out_ggga(x1, x2, x3, x4) = app3_b1_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
app19_out_gga(x1, x2, x3) = app19_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6) = U6_ggga(x6)
APP3_B1_IN_GGGA(x1, x2, x3, x4) = APP3_B1_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x4)
APP7_IN_GGA(x1, x2, x3) = APP7_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x6)
APP19_IN_GGA(x1, x2, x3) = APP19_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x5)
U5_GGGA(x1, x2, x3, x4, x5, x6) = U5_GGGA(x1, x2, x6)
U6_GGGA(x1, x2, x3, x4, x5, x6) = U6_GGGA(x6)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APP3_B1_IN_GGGA(T9, [], T18, T13) → U3_GGGA(T9, T18, T13, app7_in_gga(T9, T18, T13))
APP3_B1_IN_GGGA(T9, [], T18, T13) → APP7_IN_GGA(T9, T18, T13)
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U4_GGGA(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → APP19_IN_GGA(T48, T49, X56)
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → U2_GGA(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U5_GGGA(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_GGGA(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_GGGA(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U5_GGGA(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → APP7_IN_GGA(T9, .(T47, T52), T13)
The TRS R consists of the following rules:
app3_b1_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, app7_in_gga(T9, T18, T13))
app7_in_gga([], T25, T25) → app7_out_gga([], T25, T25)
app7_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, app7_out_gga(T35, T36, T38)) → app7_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, app7_out_gga(T9, T18, T13)) → app3_b1_out_ggga(T9, [], T18, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
app19_in_gga([], T59, T59) → app19_out_gga([], T59, T59)
app19_in_gga(.(T66, T67), T68, .(T66, X85)) → U2_gga(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
U2_gga(T66, T67, T68, X85, app19_out_gga(T67, T68, X85)) → app19_out_gga(.(T66, T67), T68, .(T66, X85))
U4_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, X56)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, app7_out_gga(T9, .(T47, T52), T13)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
The argument filtering Pi contains the following mapping:
app3_b1_in_ggga(x1, x2, x3, x4) = app3_b1_in_ggga(x1, x2, x3)
[] = []
U3_ggga(x1, x2, x3, x4) = U3_ggga(x4)
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
app7_out_gga(x1, x2, x3) = app7_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app3_b1_out_ggga(x1, x2, x3, x4) = app3_b1_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
app19_out_gga(x1, x2, x3) = app19_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6) = U6_ggga(x6)
APP3_B1_IN_GGGA(x1, x2, x3, x4) = APP3_B1_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x4)
APP7_IN_GGA(x1, x2, x3) = APP7_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x6)
APP19_IN_GGA(x1, x2, x3) = APP19_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x5)
U5_GGGA(x1, x2, x3, x4, x5, x6) = U5_GGGA(x1, x2, x6)
U6_GGGA(x1, x2, x3, x4, x5, x6) = U6_GGGA(x6)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.(8) Complex Obligation (AND)
(9) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
The TRS R consists of the following rules:
app3_b1_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, app7_in_gga(T9, T18, T13))
app7_in_gga([], T25, T25) → app7_out_gga([], T25, T25)
app7_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, app7_out_gga(T35, T36, T38)) → app7_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, app7_out_gga(T9, T18, T13)) → app3_b1_out_ggga(T9, [], T18, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
app19_in_gga([], T59, T59) → app19_out_gga([], T59, T59)
app19_in_gga(.(T66, T67), T68, .(T66, X85)) → U2_gga(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
U2_gga(T66, T67, T68, X85, app19_out_gga(T67, T68, X85)) → app19_out_gga(.(T66, T67), T68, .(T66, X85))
U4_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, X56)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, app7_out_gga(T9, .(T47, T52), T13)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
The argument filtering Pi contains the following mapping:
app3_b1_in_ggga(x1, x2, x3, x4) = app3_b1_in_ggga(x1, x2, x3)
[] = []
U3_ggga(x1, x2, x3, x4) = U3_ggga(x4)
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
app7_out_gga(x1, x2, x3) = app7_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app3_b1_out_ggga(x1, x2, x3, x4) = app3_b1_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
app19_out_gga(x1, x2, x3) = app19_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6) = U6_ggga(x6)
APP19_IN_GGA(x1, x2, x3) = APP19_IN_GGA(x1, x2)
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:
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP19_IN_GGA(x1, x2, x3) = APP19_IN_GGA(x1, x2)
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:
APP19_IN_GGA(.(T66, T67), T68) → APP19_IN_GGA(T67, T68)
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:
- APP19_IN_GGA(.(T66, T67), T68) → APP19_IN_GGA(T67, T68)
The graph contains the following edges 1 > 1, 2 >= 2
(15) YES
(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
The TRS R consists of the following rules:
app3_b1_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, app7_in_gga(T9, T18, T13))
app7_in_gga([], T25, T25) → app7_out_gga([], T25, T25)
app7_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, app7_out_gga(T35, T36, T38)) → app7_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, app7_out_gga(T9, T18, T13)) → app3_b1_out_ggga(T9, [], T18, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
app19_in_gga([], T59, T59) → app19_out_gga([], T59, T59)
app19_in_gga(.(T66, T67), T68, .(T66, X85)) → U2_gga(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
U2_gga(T66, T67, T68, X85, app19_out_gga(T67, T68, X85)) → app19_out_gga(.(T66, T67), T68, .(T66, X85))
U4_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, X56)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
app3_b1_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, app19_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, app7_out_gga(T9, .(T47, T52), T13)) → app3_b1_out_ggga(T9, .(T47, T48), T49, T13)
The argument filtering Pi contains the following mapping:
app3_b1_in_ggga(x1, x2, x3, x4) = app3_b1_in_ggga(x1, x2, x3)
[] = []
U3_ggga(x1, x2, x3, x4) = U3_ggga(x4)
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
app7_out_gga(x1, x2, x3) = app7_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app3_b1_out_ggga(x1, x2, x3, x4) = app3_b1_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
app19_out_gga(x1, x2, x3) = app19_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6) = U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6) = U6_ggga(x6)
APP7_IN_GGA(x1, x2, x3) = APP7_IN_GGA(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:
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP7_IN_GGA(x1, x2, x3) = APP7_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
(19) PiDPToQDPProof (SOUND 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:
APP7_IN_GGA(.(T34, T35), T36) → APP7_IN_GGA(T35, T36)
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:
- APP7_IN_GGA(.(T34, T35), T36) → APP7_IN_GGA(T35, T36)
The graph contains the following edges 1 > 1, 2 >= 2