/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern maximum(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) UndefinedPredicateHandlerProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 1 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 5 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Clauses: maximum(.(X, Xs), M) :- maximum(Xs, X, M). maximum(.(X, Xs), Y, M) :- ','(=<(X, Y), maximum(Xs, Y, M)). maximum(.(X, Xs), Y, M) :- ','(>(X, Y), maximum(Xs, X, M)). maximum([], M, M). Query: maximum(g,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: maximum(.(X, Xs), M) :- maximum(Xs, X, M). maximum(.(X, Xs), Y, M) :- ','(=<(X, Y), maximum(Xs, Y, M)). maximum(.(X, Xs), Y, M) :- ','(>(X, Y), maximum(Xs, X, M)). maximum([], M, M). =<(X0, X1). >(X0, X1). Query: maximum(g,a) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: maximum_in_2: (b,f) maximum_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: maximum_in_ga(.(X, Xs), M) -> U1_ga(X, Xs, M, maximum_in_gga(Xs, X, M)) maximum_in_gga(.(X, Xs), Y, M) -> U2_gga(X, Xs, Y, M, =<_in_gg(X, Y)) =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) U2_gga(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_gga(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) maximum_in_gga(.(X, Xs), Y, M) -> U4_gga(X, Xs, Y, M, >_in_gg(X, Y)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U4_gga(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_gga(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) maximum_in_gga([], M, M) -> maximum_out_gga([], M, M) U5_gga(X, Xs, Y, M, maximum_out_gga(Xs, X, M)) -> maximum_out_gga(.(X, Xs), Y, M) U3_gga(X, Xs, Y, M, maximum_out_gga(Xs, Y, M)) -> maximum_out_gga(.(X, Xs), Y, M) U1_ga(X, Xs, M, maximum_out_gga(Xs, X, M)) -> maximum_out_ga(.(X, Xs), M) The argument filtering Pi contains the following mapping: maximum_in_ga(x1, x2) = maximum_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) maximum_in_gga(x1, x2, x3) = maximum_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x2, x3, x5) =<_in_gg(x1, x2) = =<_in_gg(x1, x2) =<_out_gg(x1, x2) = =<_out_gg(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) [] = [] maximum_out_gga(x1, x2, x3) = maximum_out_gga(x1, x2, x3) maximum_out_ga(x1, x2) = maximum_out_ga(x1, 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: maximum_in_ga(.(X, Xs), M) -> U1_ga(X, Xs, M, maximum_in_gga(Xs, X, M)) maximum_in_gga(.(X, Xs), Y, M) -> U2_gga(X, Xs, Y, M, =<_in_gg(X, Y)) =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) U2_gga(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_gga(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) maximum_in_gga(.(X, Xs), Y, M) -> U4_gga(X, Xs, Y, M, >_in_gg(X, Y)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U4_gga(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_gga(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) maximum_in_gga([], M, M) -> maximum_out_gga([], M, M) U5_gga(X, Xs, Y, M, maximum_out_gga(Xs, X, M)) -> maximum_out_gga(.(X, Xs), Y, M) U3_gga(X, Xs, Y, M, maximum_out_gga(Xs, Y, M)) -> maximum_out_gga(.(X, Xs), Y, M) U1_ga(X, Xs, M, maximum_out_gga(Xs, X, M)) -> maximum_out_ga(.(X, Xs), M) The argument filtering Pi contains the following mapping: maximum_in_ga(x1, x2) = maximum_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) maximum_in_gga(x1, x2, x3) = maximum_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x2, x3, x5) =<_in_gg(x1, x2) = =<_in_gg(x1, x2) =<_out_gg(x1, x2) = =<_out_gg(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) [] = [] maximum_out_gga(x1, x2, x3) = maximum_out_gga(x1, x2, x3) maximum_out_ga(x1, x2) = maximum_out_ga(x1, x2) ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: MAXIMUM_IN_GA(.(X, Xs), M) -> U1_GA(X, Xs, M, maximum_in_gga(Xs, X, M)) MAXIMUM_IN_GA(.(X, Xs), M) -> MAXIMUM_IN_GGA(Xs, X, M) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U2_GGA(X, Xs, Y, M, =<_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> =<_IN_GG(X, Y) U2_GGA(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_GGA(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) U2_GGA(X, Xs, Y, M, =<_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, Y, M) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U4_GGA(X, Xs, Y, M, >_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> >_IN_GG(X, Y) U4_GGA(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_GGA(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) U4_GGA(X, Xs, Y, M, >_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, X, M) The TRS R consists of the following rules: maximum_in_ga(.(X, Xs), M) -> U1_ga(X, Xs, M, maximum_in_gga(Xs, X, M)) maximum_in_gga(.(X, Xs), Y, M) -> U2_gga(X, Xs, Y, M, =<_in_gg(X, Y)) =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) U2_gga(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_gga(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) maximum_in_gga(.(X, Xs), Y, M) -> U4_gga(X, Xs, Y, M, >_in_gg(X, Y)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U4_gga(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_gga(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) maximum_in_gga([], M, M) -> maximum_out_gga([], M, M) U5_gga(X, Xs, Y, M, maximum_out_gga(Xs, X, M)) -> maximum_out_gga(.(X, Xs), Y, M) U3_gga(X, Xs, Y, M, maximum_out_gga(Xs, Y, M)) -> maximum_out_gga(.(X, Xs), Y, M) U1_ga(X, Xs, M, maximum_out_gga(Xs, X, M)) -> maximum_out_ga(.(X, Xs), M) The argument filtering Pi contains the following mapping: maximum_in_ga(x1, x2) = maximum_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) maximum_in_gga(x1, x2, x3) = maximum_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x2, x3, x5) =<_in_gg(x1, x2) = =<_in_gg(x1, x2) =<_out_gg(x1, x2) = =<_out_gg(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) [] = [] maximum_out_gga(x1, x2, x3) = maximum_out_gga(x1, x2, x3) maximum_out_ga(x1, x2) = maximum_out_ga(x1, x2) MAXIMUM_IN_GA(x1, x2) = MAXIMUM_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) MAXIMUM_IN_GGA(x1, x2, x3) = MAXIMUM_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) =<_IN_GG(x1, x2) = =<_IN_GG(x1, x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: MAXIMUM_IN_GA(.(X, Xs), M) -> U1_GA(X, Xs, M, maximum_in_gga(Xs, X, M)) MAXIMUM_IN_GA(.(X, Xs), M) -> MAXIMUM_IN_GGA(Xs, X, M) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U2_GGA(X, Xs, Y, M, =<_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> =<_IN_GG(X, Y) U2_GGA(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_GGA(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) U2_GGA(X, Xs, Y, M, =<_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, Y, M) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U4_GGA(X, Xs, Y, M, >_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> >_IN_GG(X, Y) U4_GGA(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_GGA(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) U4_GGA(X, Xs, Y, M, >_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, X, M) The TRS R consists of the following rules: maximum_in_ga(.(X, Xs), M) -> U1_ga(X, Xs, M, maximum_in_gga(Xs, X, M)) maximum_in_gga(.(X, Xs), Y, M) -> U2_gga(X, Xs, Y, M, =<_in_gg(X, Y)) =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) U2_gga(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_gga(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) maximum_in_gga(.(X, Xs), Y, M) -> U4_gga(X, Xs, Y, M, >_in_gg(X, Y)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U4_gga(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_gga(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) maximum_in_gga([], M, M) -> maximum_out_gga([], M, M) U5_gga(X, Xs, Y, M, maximum_out_gga(Xs, X, M)) -> maximum_out_gga(.(X, Xs), Y, M) U3_gga(X, Xs, Y, M, maximum_out_gga(Xs, Y, M)) -> maximum_out_gga(.(X, Xs), Y, M) U1_ga(X, Xs, M, maximum_out_gga(Xs, X, M)) -> maximum_out_ga(.(X, Xs), M) The argument filtering Pi contains the following mapping: maximum_in_ga(x1, x2) = maximum_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) maximum_in_gga(x1, x2, x3) = maximum_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x2, x3, x5) =<_in_gg(x1, x2) = =<_in_gg(x1, x2) =<_out_gg(x1, x2) = =<_out_gg(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) [] = [] maximum_out_gga(x1, x2, x3) = maximum_out_gga(x1, x2, x3) maximum_out_ga(x1, x2) = maximum_out_ga(x1, x2) MAXIMUM_IN_GA(x1, x2) = MAXIMUM_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) MAXIMUM_IN_GGA(x1, x2, x3) = MAXIMUM_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) =<_IN_GG(x1, x2) = =<_IN_GG(x1, x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GGA(X, Xs, Y, M, =<_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, Y, M) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U2_GGA(X, Xs, Y, M, =<_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U4_GGA(X, Xs, Y, M, >_in_gg(X, Y)) U4_GGA(X, Xs, Y, M, >_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, X, M) The TRS R consists of the following rules: maximum_in_ga(.(X, Xs), M) -> U1_ga(X, Xs, M, maximum_in_gga(Xs, X, M)) maximum_in_gga(.(X, Xs), Y, M) -> U2_gga(X, Xs, Y, M, =<_in_gg(X, Y)) =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) U2_gga(X, Xs, Y, M, =<_out_gg(X, Y)) -> U3_gga(X, Xs, Y, M, maximum_in_gga(Xs, Y, M)) maximum_in_gga(.(X, Xs), Y, M) -> U4_gga(X, Xs, Y, M, >_in_gg(X, Y)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U4_gga(X, Xs, Y, M, >_out_gg(X, Y)) -> U5_gga(X, Xs, Y, M, maximum_in_gga(Xs, X, M)) maximum_in_gga([], M, M) -> maximum_out_gga([], M, M) U5_gga(X, Xs, Y, M, maximum_out_gga(Xs, X, M)) -> maximum_out_gga(.(X, Xs), Y, M) U3_gga(X, Xs, Y, M, maximum_out_gga(Xs, Y, M)) -> maximum_out_gga(.(X, Xs), Y, M) U1_ga(X, Xs, M, maximum_out_gga(Xs, X, M)) -> maximum_out_ga(.(X, Xs), M) The argument filtering Pi contains the following mapping: maximum_in_ga(x1, x2) = maximum_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) maximum_in_gga(x1, x2, x3) = maximum_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x2, x3, x5) =<_in_gg(x1, x2) = =<_in_gg(x1, x2) =<_out_gg(x1, x2) = =<_out_gg(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) [] = [] maximum_out_gga(x1, x2, x3) = maximum_out_gga(x1, x2, x3) maximum_out_ga(x1, x2) = maximum_out_ga(x1, x2) MAXIMUM_IN_GGA(x1, x2, x3) = MAXIMUM_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GGA(X, Xs, Y, M, =<_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, Y, M) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U2_GGA(X, Xs, Y, M, =<_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y, M) -> U4_GGA(X, Xs, Y, M, >_in_gg(X, Y)) U4_GGA(X, Xs, Y, M, >_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, X, M) The TRS R consists of the following rules: =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) >_in_gg(X0, X1) -> >_out_gg(X0, X1) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) =<_in_gg(x1, x2) = =<_in_gg(x1, x2) =<_out_gg(x1, x2) = =<_out_gg(x1, x2) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg(x1, x2) MAXIMUM_IN_GGA(x1, x2, x3) = MAXIMUM_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GGA(X, Xs, Y, =<_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, Y) MAXIMUM_IN_GGA(.(X, Xs), Y) -> U2_GGA(X, Xs, Y, =<_in_gg(X, Y)) MAXIMUM_IN_GGA(.(X, Xs), Y) -> U4_GGA(X, Xs, Y, >_in_gg(X, Y)) U4_GGA(X, Xs, Y, >_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, X) The TRS R consists of the following rules: =<_in_gg(X0, X1) -> =<_out_gg(X0, X1) >_in_gg(X0, X1) -> >_out_gg(X0, X1) The set Q consists of the following terms: =<_in_gg(x0, x1) >_in_gg(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) QDPSizeChangeProof (EQUIVALENT) 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: *MAXIMUM_IN_GGA(.(X, Xs), Y) -> U2_GGA(X, Xs, Y, =<_in_gg(X, Y)) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3 *MAXIMUM_IN_GGA(.(X, Xs), Y) -> U4_GGA(X, Xs, Y, >_in_gg(X, Y)) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3 *U2_GGA(X, Xs, Y, =<_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, Y) The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2 *U4_GGA(X, Xs, Y, >_out_gg(X, Y)) -> MAXIMUM_IN_GGA(Xs, X) The graph contains the following edges 2 >= 1, 1 >= 2, 4 > 2 ---------------------------------------- (14) YES