5.21/2.57 YES 5.21/2.59 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 5.21/2.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.21/2.59 5.21/2.59 5.21/2.59 Left Termination of the query pattern 5.21/2.59 5.21/2.59 log2(g,a) 5.21/2.59 5.21/2.59 w.r.t. the given Prolog program could successfully be proven: 5.21/2.59 5.21/2.59 (0) Prolog 5.21/2.59 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.21/2.59 (2) PiTRS 5.21/2.59 (3) DependencyPairsProof [EQUIVALENT, 7 ms] 5.21/2.59 (4) PiDP 5.21/2.59 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.21/2.59 (6) PiDP 5.21/2.59 (7) UsableRulesProof [EQUIVALENT, 0 ms] 5.21/2.59 (8) PiDP 5.21/2.59 (9) PiDPToQDPProof [SOUND, 0 ms] 5.21/2.59 (10) QDP 5.21/2.59 (11) MRRProof [EQUIVALENT, 73 ms] 5.21/2.59 (12) QDP 5.21/2.59 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 5.21/2.59 (14) TRUE 5.21/2.59 5.21/2.59 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (0) 5.21/2.59 Obligation: 5.21/2.59 Clauses: 5.21/2.59 5.21/2.59 log2(X, Y) :- log2(X, 0, s(0), Y). 5.21/2.59 log2(s(s(X)), Half, Acc, Y) :- log2(X, s(Half), Acc, Y). 5.21/2.59 log2(X, s(s(Half)), Acc, Y) :- ','(small(X), log2(Half, s(0), s(Acc), Y)). 5.21/2.59 log2(X, Half, Y, Y) :- ','(small(X), small(Half)). 5.21/2.59 small(0). 5.21/2.59 small(s(0)). 5.21/2.59 5.21/2.59 5.21/2.59 Query: log2(g,a) 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (1) PrologToPiTRSProof (SOUND) 5.21/2.59 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.21/2.59 5.21/2.59 log2_in_2: (b,f) 5.21/2.59 5.21/2.59 log2_in_4: (b,b,b,f) 5.21/2.59 5.21/2.59 Transforming Prolog into the following Term Rewriting System: 5.21/2.59 5.21/2.59 Pi-finite rewrite system: 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 small_in_g(0) -> small_out_g(0) 5.21/2.59 small_in_g(s(0)) -> small_out_g(s(0)) 5.21/2.59 U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X)) 5.21/2.59 U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half)) 5.21/2.59 U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y) 5.21/2.59 U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y) 5.21/2.59 U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y) 5.21/2.59 U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y) 5.21/2.59 5.21/2.59 The argument filtering Pi contains the following mapping: 5.21/2.59 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.21/2.59 5.21/2.59 U1_ga(x1, x2, x3) = U1_ga(x3) 5.21/2.59 5.21/2.59 log2_in_ggga(x1, x2, x3, x4) = log2_in_ggga(x1, x2, x3) 5.21/2.59 5.21/2.59 s(x1) = s(x1) 5.21/2.59 5.21/2.59 U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) 5.21/2.59 5.21/2.59 U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x2, x3, x5) 5.21/2.59 5.21/2.59 small_in_g(x1) = small_in_g(x1) 5.21/2.59 5.21/2.59 0 = 0 5.21/2.59 5.21/2.59 small_out_g(x1) = small_out_g 5.21/2.59 5.21/2.59 U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x5) 5.21/2.59 5.21/2.59 U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_ggga(x1, x2, x3, x4) = U6_ggga(x3, x4) 5.21/2.59 5.21/2.59 log2_out_ggga(x1, x2, x3, x4) = log2_out_ggga(x4) 5.21/2.59 5.21/2.59 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.21/2.59 5.21/2.59 5.21/2.59 5.21/2.59 5.21/2.59 5.21/2.59 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.21/2.59 5.21/2.59 5.21/2.59 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (2) 5.21/2.59 Obligation: 5.21/2.59 Pi-finite rewrite system: 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 small_in_g(0) -> small_out_g(0) 5.21/2.59 small_in_g(s(0)) -> small_out_g(s(0)) 5.21/2.59 U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X)) 5.21/2.59 U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half)) 5.21/2.59 U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y) 5.21/2.59 U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y) 5.21/2.59 U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y) 5.21/2.59 U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y) 5.21/2.59 5.21/2.59 The argument filtering Pi contains the following mapping: 5.21/2.59 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.21/2.59 5.21/2.59 U1_ga(x1, x2, x3) = U1_ga(x3) 5.21/2.59 5.21/2.59 log2_in_ggga(x1, x2, x3, x4) = log2_in_ggga(x1, x2, x3) 5.21/2.59 5.21/2.59 s(x1) = s(x1) 5.21/2.59 5.21/2.59 U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) 5.21/2.59 5.21/2.59 U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x2, x3, x5) 5.21/2.59 5.21/2.59 small_in_g(x1) = small_in_g(x1) 5.21/2.59 5.21/2.59 0 = 0 5.21/2.59 5.21/2.59 small_out_g(x1) = small_out_g 5.21/2.59 5.21/2.59 U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x5) 5.21/2.59 5.21/2.59 U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_ggga(x1, x2, x3, x4) = U6_ggga(x3, x4) 5.21/2.59 5.21/2.59 log2_out_ggga(x1, x2, x3, x4) = log2_out_ggga(x4) 5.21/2.59 5.21/2.59 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.21/2.59 5.21/2.59 5.21/2.59 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (3) DependencyPairsProof (EQUIVALENT) 5.21/2.59 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.21/2.59 Pi DP problem: 5.21/2.59 The TRS P consists of the following rules: 5.21/2.59 5.21/2.59 LOG2_IN_GA(X, Y) -> U1_GA(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 LOG2_IN_GA(X, Y) -> LOG2_IN_GGGA(X, 0, s(0), Y) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> U2_GGGA(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y) 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> SMALL_IN_G(X) 5.21/2.59 U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> U4_GGGA(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y) 5.21/2.59 LOG2_IN_GGGA(X, Half, Y, Y) -> U5_GGGA(X, Half, Y, small_in_g(X)) 5.21/2.59 LOG2_IN_GGGA(X, Half, Y, Y) -> SMALL_IN_G(X) 5.21/2.59 U5_GGGA(X, Half, Y, small_out_g(X)) -> U6_GGGA(X, Half, Y, small_in_g(Half)) 5.21/2.59 U5_GGGA(X, Half, Y, small_out_g(X)) -> SMALL_IN_G(Half) 5.21/2.59 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 small_in_g(0) -> small_out_g(0) 5.21/2.59 small_in_g(s(0)) -> small_out_g(s(0)) 5.21/2.59 U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X)) 5.21/2.59 U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half)) 5.21/2.59 U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y) 5.21/2.59 U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y) 5.21/2.59 U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y) 5.21/2.59 U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y) 5.21/2.59 5.21/2.59 The argument filtering Pi contains the following mapping: 5.21/2.59 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.21/2.59 5.21/2.59 U1_ga(x1, x2, x3) = U1_ga(x3) 5.21/2.59 5.21/2.59 log2_in_ggga(x1, x2, x3, x4) = log2_in_ggga(x1, x2, x3) 5.21/2.59 5.21/2.59 s(x1) = s(x1) 5.21/2.59 5.21/2.59 U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) 5.21/2.59 5.21/2.59 U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x2, x3, x5) 5.21/2.59 5.21/2.59 small_in_g(x1) = small_in_g(x1) 5.21/2.59 5.21/2.59 0 = 0 5.21/2.59 5.21/2.59 small_out_g(x1) = small_out_g 5.21/2.59 5.21/2.59 U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x5) 5.21/2.59 5.21/2.59 U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_ggga(x1, x2, x3, x4) = U6_ggga(x3, x4) 5.21/2.59 5.21/2.59 log2_out_ggga(x1, x2, x3, x4) = log2_out_ggga(x4) 5.21/2.59 5.21/2.59 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.21/2.59 5.21/2.59 LOG2_IN_GA(x1, x2) = LOG2_IN_GA(x1) 5.21/2.59 5.21/2.59 U1_GA(x1, x2, x3) = U1_GA(x3) 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(x1, x2, x3, x4) = LOG2_IN_GGGA(x1, x2, x3) 5.21/2.59 5.21/2.59 U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x5) 5.21/2.59 5.21/2.59 U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x2, x3, x5) 5.21/2.59 5.21/2.59 SMALL_IN_G(x1) = SMALL_IN_G(x1) 5.21/2.59 5.21/2.59 U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x5) 5.21/2.59 5.21/2.59 U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x3, x4) 5.21/2.59 5.21/2.59 5.21/2.59 We have to consider all (P,R,Pi)-chains 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (4) 5.21/2.59 Obligation: 5.21/2.59 Pi DP problem: 5.21/2.59 The TRS P consists of the following rules: 5.21/2.59 5.21/2.59 LOG2_IN_GA(X, Y) -> U1_GA(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 LOG2_IN_GA(X, Y) -> LOG2_IN_GGGA(X, 0, s(0), Y) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> U2_GGGA(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y) 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> SMALL_IN_G(X) 5.21/2.59 U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> U4_GGGA(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y) 5.21/2.59 LOG2_IN_GGGA(X, Half, Y, Y) -> U5_GGGA(X, Half, Y, small_in_g(X)) 5.21/2.59 LOG2_IN_GGGA(X, Half, Y, Y) -> SMALL_IN_G(X) 5.21/2.59 U5_GGGA(X, Half, Y, small_out_g(X)) -> U6_GGGA(X, Half, Y, small_in_g(Half)) 5.21/2.59 U5_GGGA(X, Half, Y, small_out_g(X)) -> SMALL_IN_G(Half) 5.21/2.59 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 small_in_g(0) -> small_out_g(0) 5.21/2.59 small_in_g(s(0)) -> small_out_g(s(0)) 5.21/2.59 U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X)) 5.21/2.59 U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half)) 5.21/2.59 U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y) 5.21/2.59 U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y) 5.21/2.59 U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y) 5.21/2.59 U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y) 5.21/2.59 5.21/2.59 The argument filtering Pi contains the following mapping: 5.21/2.59 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.21/2.59 5.21/2.59 U1_ga(x1, x2, x3) = U1_ga(x3) 5.21/2.59 5.21/2.59 log2_in_ggga(x1, x2, x3, x4) = log2_in_ggga(x1, x2, x3) 5.21/2.59 5.21/2.59 s(x1) = s(x1) 5.21/2.59 5.21/2.59 U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) 5.21/2.59 5.21/2.59 U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x2, x3, x5) 5.21/2.59 5.21/2.59 small_in_g(x1) = small_in_g(x1) 5.21/2.59 5.21/2.59 0 = 0 5.21/2.59 5.21/2.59 small_out_g(x1) = small_out_g 5.21/2.59 5.21/2.59 U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x5) 5.21/2.59 5.21/2.59 U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_ggga(x1, x2, x3, x4) = U6_ggga(x3, x4) 5.21/2.59 5.21/2.59 log2_out_ggga(x1, x2, x3, x4) = log2_out_ggga(x4) 5.21/2.59 5.21/2.59 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.21/2.59 5.21/2.59 LOG2_IN_GA(x1, x2) = LOG2_IN_GA(x1) 5.21/2.59 5.21/2.59 U1_GA(x1, x2, x3) = U1_GA(x3) 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(x1, x2, x3, x4) = LOG2_IN_GGGA(x1, x2, x3) 5.21/2.59 5.21/2.59 U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x5) 5.21/2.59 5.21/2.59 U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x2, x3, x5) 5.21/2.59 5.21/2.59 SMALL_IN_G(x1) = SMALL_IN_G(x1) 5.21/2.59 5.21/2.59 U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x5) 5.21/2.59 5.21/2.59 U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x3, x4) 5.21/2.59 5.21/2.59 5.21/2.59 We have to consider all (P,R,Pi)-chains 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (5) DependencyGraphProof (EQUIVALENT) 5.21/2.59 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 9 less nodes. 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (6) 5.21/2.59 Obligation: 5.21/2.59 Pi DP problem: 5.21/2.59 The TRS P consists of the following rules: 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y) 5.21/2.59 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_ggga(X, 0, s(0), Y)) 5.21/2.59 log2_in_ggga(s(s(X)), Half, Acc, Y) -> U2_ggga(X, Half, Acc, Y, log2_in_ggga(X, s(Half), Acc, Y)) 5.21/2.59 log2_in_ggga(X, s(s(Half)), Acc, Y) -> U3_ggga(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 small_in_g(0) -> small_out_g(0) 5.21/2.59 small_in_g(s(0)) -> small_out_g(s(0)) 5.21/2.59 U3_ggga(X, Half, Acc, Y, small_out_g(X)) -> U4_ggga(X, Half, Acc, Y, log2_in_ggga(Half, s(0), s(Acc), Y)) 5.21/2.59 log2_in_ggga(X, Half, Y, Y) -> U5_ggga(X, Half, Y, small_in_g(X)) 5.21/2.59 U5_ggga(X, Half, Y, small_out_g(X)) -> U6_ggga(X, Half, Y, small_in_g(Half)) 5.21/2.59 U6_ggga(X, Half, Y, small_out_g(Half)) -> log2_out_ggga(X, Half, Y, Y) 5.21/2.59 U4_ggga(X, Half, Acc, Y, log2_out_ggga(Half, s(0), s(Acc), Y)) -> log2_out_ggga(X, s(s(Half)), Acc, Y) 5.21/2.59 U2_ggga(X, Half, Acc, Y, log2_out_ggga(X, s(Half), Acc, Y)) -> log2_out_ggga(s(s(X)), Half, Acc, Y) 5.21/2.59 U1_ga(X, Y, log2_out_ggga(X, 0, s(0), Y)) -> log2_out_ga(X, Y) 5.21/2.59 5.21/2.59 The argument filtering Pi contains the following mapping: 5.21/2.59 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.21/2.59 5.21/2.59 U1_ga(x1, x2, x3) = U1_ga(x3) 5.21/2.59 5.21/2.59 log2_in_ggga(x1, x2, x3, x4) = log2_in_ggga(x1, x2, x3) 5.21/2.59 5.21/2.59 s(x1) = s(x1) 5.21/2.59 5.21/2.59 U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) 5.21/2.59 5.21/2.59 U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x2, x3, x5) 5.21/2.59 5.21/2.59 small_in_g(x1) = small_in_g(x1) 5.21/2.59 5.21/2.59 0 = 0 5.21/2.59 5.21/2.59 small_out_g(x1) = small_out_g 5.21/2.59 5.21/2.59 U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x5) 5.21/2.59 5.21/2.59 U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x3, x4) 5.21/2.59 5.21/2.59 U6_ggga(x1, x2, x3, x4) = U6_ggga(x3, x4) 5.21/2.59 5.21/2.59 log2_out_ggga(x1, x2, x3, x4) = log2_out_ggga(x4) 5.21/2.59 5.21/2.59 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(x1, x2, x3, x4) = LOG2_IN_GGGA(x1, x2, x3) 5.21/2.59 5.21/2.59 U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x2, x3, x5) 5.21/2.59 5.21/2.59 5.21/2.59 We have to consider all (P,R,Pi)-chains 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (7) UsableRulesProof (EQUIVALENT) 5.21/2.59 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (8) 5.21/2.59 Obligation: 5.21/2.59 Pi DP problem: 5.21/2.59 The TRS P consists of the following rules: 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc, Y) -> U3_GGGA(X, Half, Acc, Y, small_in_g(X)) 5.21/2.59 U3_GGGA(X, Half, Acc, Y, small_out_g(X)) -> LOG2_IN_GGGA(Half, s(0), s(Acc), Y) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc, Y) -> LOG2_IN_GGGA(X, s(Half), Acc, Y) 5.21/2.59 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 small_in_g(0) -> small_out_g(0) 5.21/2.59 small_in_g(s(0)) -> small_out_g(s(0)) 5.21/2.59 5.21/2.59 The argument filtering Pi contains the following mapping: 5.21/2.59 s(x1) = s(x1) 5.21/2.59 5.21/2.59 small_in_g(x1) = small_in_g(x1) 5.21/2.59 5.21/2.59 0 = 0 5.21/2.59 5.21/2.59 small_out_g(x1) = small_out_g 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(x1, x2, x3, x4) = LOG2_IN_GGGA(x1, x2, x3) 5.21/2.59 5.21/2.59 U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x2, x3, x5) 5.21/2.59 5.21/2.59 5.21/2.59 We have to consider all (P,R,Pi)-chains 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (9) PiDPToQDPProof (SOUND) 5.21/2.59 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (10) 5.21/2.59 Obligation: 5.21/2.59 Q DP problem: 5.21/2.59 The TRS P consists of the following rules: 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc) -> U3_GGGA(Half, Acc, small_in_g(X)) 5.21/2.59 U3_GGGA(Half, Acc, small_out_g) -> LOG2_IN_GGGA(Half, s(0), s(Acc)) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc) -> LOG2_IN_GGGA(X, s(Half), Acc) 5.21/2.59 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 small_in_g(0) -> small_out_g 5.21/2.59 small_in_g(s(0)) -> small_out_g 5.21/2.59 5.21/2.59 The set Q consists of the following terms: 5.21/2.59 5.21/2.59 small_in_g(x0) 5.21/2.59 5.21/2.59 We have to consider all (P,Q,R)-chains. 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (11) MRRProof (EQUIVALENT) 5.21/2.59 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 5.21/2.59 5.21/2.59 Strictly oriented dependency pairs: 5.21/2.59 5.21/2.59 U3_GGGA(Half, Acc, small_out_g) -> LOG2_IN_GGGA(Half, s(0), s(Acc)) 5.21/2.59 LOG2_IN_GGGA(s(s(X)), Half, Acc) -> LOG2_IN_GGGA(X, s(Half), Acc) 5.21/2.59 5.21/2.59 Strictly oriented rules of the TRS R: 5.21/2.59 5.21/2.59 small_in_g(s(0)) -> small_out_g 5.21/2.59 5.21/2.59 Used ordering: Polynomial interpretation [POLO]: 5.21/2.59 5.21/2.59 POL(0) = 0 5.21/2.59 POL(LOG2_IN_GGGA(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 5.21/2.59 POL(U3_GGGA(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 5.21/2.59 POL(s(x_1)) = 1 + x_1 5.21/2.59 POL(small_in_g(x_1)) = 2 + x_1 5.21/2.59 POL(small_out_g) = 2 5.21/2.59 5.21/2.59 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (12) 5.21/2.59 Obligation: 5.21/2.59 Q DP problem: 5.21/2.59 The TRS P consists of the following rules: 5.21/2.59 5.21/2.59 LOG2_IN_GGGA(X, s(s(Half)), Acc) -> U3_GGGA(Half, Acc, small_in_g(X)) 5.21/2.59 5.21/2.59 The TRS R consists of the following rules: 5.21/2.59 5.21/2.59 small_in_g(0) -> small_out_g 5.21/2.59 5.21/2.59 The set Q consists of the following terms: 5.21/2.59 5.21/2.59 small_in_g(x0) 5.21/2.59 5.21/2.59 We have to consider all (P,Q,R)-chains. 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (13) DependencyGraphProof (EQUIVALENT) 5.21/2.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.21/2.59 ---------------------------------------- 5.21/2.59 5.21/2.59 (14) 5.21/2.59 TRUE 5.43/2.64 EOF