8.66/3.53 YES 8.66/3.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 8.66/3.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.66/3.56 8.66/3.56 8.66/3.56 Left Termination of the query pattern 8.66/3.56 8.66/3.56 rev(g,a) 8.66/3.56 8.66/3.56 w.r.t. the given Prolog program could successfully be proven: 8.66/3.56 8.66/3.56 (0) Prolog 8.66/3.56 (1) PrologToPiTRSProof [SOUND, 0 ms] 8.66/3.56 (2) PiTRS 8.66/3.56 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 8.66/3.56 (4) PiDP 8.66/3.56 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 8.66/3.56 (6) AND 8.66/3.56 (7) PiDP 8.66/3.56 (8) UsableRulesProof [EQUIVALENT, 0 ms] 8.66/3.56 (9) PiDP 8.66/3.56 (10) PiDPToQDPProof [SOUND, 1 ms] 8.66/3.56 (11) QDP 8.66/3.56 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 8.66/3.56 (13) YES 8.66/3.56 (14) PiDP 8.66/3.56 (15) PiDPToQDPProof [SOUND, 1 ms] 8.66/3.56 (16) QDP 8.66/3.56 (17) QDPQMonotonicMRRProof [EQUIVALENT, 71 ms] 8.66/3.56 (18) QDP 8.66/3.56 (19) DependencyGraphProof [EQUIVALENT, 0 ms] 8.66/3.56 (20) TRUE 8.66/3.56 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (0) 8.66/3.56 Obligation: 8.66/3.56 Clauses: 8.66/3.56 8.66/3.56 rev([], []). 8.66/3.56 rev(.(X, XS), .(Y, YS)) :- ','(rev1(X, XS, Y), rev2(X, XS, YS)). 8.66/3.56 rev1(X, [], X). 8.66/3.56 rev1(X, .(Y, YS), Z) :- rev1(Y, YS, Z). 8.66/3.56 rev2(X, [], []). 8.66/3.56 rev2(X, .(Y, YS), ZS) :- ','(rev2(Y, YS, US), ','(rev(US, VS), rev(.(X, VS), ZS))). 8.66/3.56 8.66/3.56 8.66/3.56 Query: rev(g,a) 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (1) PrologToPiTRSProof (SOUND) 8.66/3.56 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 8.66/3.56 8.66/3.56 rev_in_2: (b,f) 8.66/3.56 8.66/3.56 rev1_in_3: (b,b,f) 8.66/3.56 8.66/3.56 rev2_in_3: (b,b,f) 8.66/3.56 8.66/3.56 Transforming Prolog into the following Term Rewriting System: 8.66/3.56 8.66/3.56 Pi-finite rewrite system: 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([], []) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS), .(Y, YS)) -> U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 rev1_in_gga(X, [], X) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS), Z) -> U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 rev2_in_gga(X, [], []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS), ZS) -> U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 rev_in_ga(x1, x2) = rev_in_ga(x1) 8.66/3.56 8.66/3.56 [] = [] 8.66/3.56 8.66/3.56 rev_out_ga(x1, x2) = rev_out_ga(x1, x2) 8.66/3.56 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) 8.66/3.56 8.66/3.56 rev1_in_gga(x1, x2, x3) = rev1_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev1_out_gga(x1, x2, x3) = rev1_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 rev2_in_gga(x1, x2, x3) = rev2_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev2_out_gga(x1, x2, x3) = rev2_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 8.66/3.56 8.66/3.56 8.66/3.56 8.66/3.56 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 8.66/3.56 8.66/3.56 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (2) 8.66/3.56 Obligation: 8.66/3.56 Pi-finite rewrite system: 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([], []) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS), .(Y, YS)) -> U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 rev1_in_gga(X, [], X) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS), Z) -> U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 rev2_in_gga(X, [], []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS), ZS) -> U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 rev_in_ga(x1, x2) = rev_in_ga(x1) 8.66/3.56 8.66/3.56 [] = [] 8.66/3.56 8.66/3.56 rev_out_ga(x1, x2) = rev_out_ga(x1, x2) 8.66/3.56 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) 8.66/3.56 8.66/3.56 rev1_in_gga(x1, x2, x3) = rev1_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev1_out_gga(x1, x2, x3) = rev1_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 rev2_in_gga(x1, x2, x3) = rev2_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev2_out_gga(x1, x2, x3) = rev2_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (3) DependencyPairsProof (EQUIVALENT) 8.66/3.56 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 8.66/3.56 Pi DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 REV_IN_GA(.(X, XS), .(Y, YS)) -> U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 REV_IN_GA(.(X, XS), .(Y, YS)) -> REV1_IN_GGA(X, XS, Y) 8.66/3.56 REV1_IN_GGA(X, .(Y, YS), Z) -> U3_GGA(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 REV1_IN_GGA(X, .(Y, YS), Z) -> REV1_IN_GGA(Y, YS, Z) 8.66/3.56 U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_GA(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> REV2_IN_GGA(X, XS, YS) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS), ZS) -> U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS), ZS) -> REV2_IN_GGA(Y, YS, US) 8.66/3.56 U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> REV_IN_GA(US, VS) 8.66/3.56 U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_GGA(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) -> REV_IN_GA(.(X, VS), ZS) 8.66/3.56 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([], []) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS), .(Y, YS)) -> U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 rev1_in_gga(X, [], X) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS), Z) -> U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 rev2_in_gga(X, [], []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS), ZS) -> U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 rev_in_ga(x1, x2) = rev_in_ga(x1) 8.66/3.56 8.66/3.56 [] = [] 8.66/3.56 8.66/3.56 rev_out_ga(x1, x2) = rev_out_ga(x1, x2) 8.66/3.56 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) 8.66/3.56 8.66/3.56 rev1_in_gga(x1, x2, x3) = rev1_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev1_out_gga(x1, x2, x3) = rev1_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 rev2_in_gga(x1, x2, x3) = rev2_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev2_out_gga(x1, x2, x3) = rev2_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 REV_IN_GA(x1, x2) = REV_IN_GA(x1) 8.66/3.56 8.66/3.56 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) 8.66/3.56 8.66/3.56 REV1_IN_GGA(x1, x2, x3) = REV1_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 REV2_IN_GGA(x1, x2, x3) = REV2_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 8.66/3.56 We have to consider all (P,R,Pi)-chains 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (4) 8.66/3.56 Obligation: 8.66/3.56 Pi DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 REV_IN_GA(.(X, XS), .(Y, YS)) -> U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 REV_IN_GA(.(X, XS), .(Y, YS)) -> REV1_IN_GGA(X, XS, Y) 8.66/3.56 REV1_IN_GGA(X, .(Y, YS), Z) -> U3_GGA(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 REV1_IN_GGA(X, .(Y, YS), Z) -> REV1_IN_GGA(Y, YS, Z) 8.66/3.56 U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_GA(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> REV2_IN_GGA(X, XS, YS) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS), ZS) -> U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS), ZS) -> REV2_IN_GGA(Y, YS, US) 8.66/3.56 U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> REV_IN_GA(US, VS) 8.66/3.56 U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_GGA(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) -> REV_IN_GA(.(X, VS), ZS) 8.66/3.56 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([], []) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS), .(Y, YS)) -> U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 rev1_in_gga(X, [], X) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS), Z) -> U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 rev2_in_gga(X, [], []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS), ZS) -> U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 rev_in_ga(x1, x2) = rev_in_ga(x1) 8.66/3.56 8.66/3.56 [] = [] 8.66/3.56 8.66/3.56 rev_out_ga(x1, x2) = rev_out_ga(x1, x2) 8.66/3.56 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) 8.66/3.56 8.66/3.56 rev1_in_gga(x1, x2, x3) = rev1_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev1_out_gga(x1, x2, x3) = rev1_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 rev2_in_gga(x1, x2, x3) = rev2_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev2_out_gga(x1, x2, x3) = rev2_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 REV_IN_GA(x1, x2) = REV_IN_GA(x1) 8.66/3.56 8.66/3.56 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) 8.66/3.56 8.66/3.56 REV1_IN_GGA(x1, x2, x3) = REV1_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 REV2_IN_GGA(x1, x2, x3) = REV2_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 8.66/3.56 We have to consider all (P,R,Pi)-chains 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (5) DependencyGraphProof (EQUIVALENT) 8.66/3.56 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (6) 8.66/3.56 Complex Obligation (AND) 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (7) 8.66/3.56 Obligation: 8.66/3.56 Pi DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 REV1_IN_GGA(X, .(Y, YS), Z) -> REV1_IN_GGA(Y, YS, Z) 8.66/3.56 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([], []) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS), .(Y, YS)) -> U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 rev1_in_gga(X, [], X) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS), Z) -> U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 rev2_in_gga(X, [], []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS), ZS) -> U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 rev_in_ga(x1, x2) = rev_in_ga(x1) 8.66/3.56 8.66/3.56 [] = [] 8.66/3.56 8.66/3.56 rev_out_ga(x1, x2) = rev_out_ga(x1, x2) 8.66/3.56 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) 8.66/3.56 8.66/3.56 rev1_in_gga(x1, x2, x3) = rev1_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev1_out_gga(x1, x2, x3) = rev1_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 rev2_in_gga(x1, x2, x3) = rev2_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev2_out_gga(x1, x2, x3) = rev2_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 REV1_IN_GGA(x1, x2, x3) = REV1_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 8.66/3.56 We have to consider all (P,R,Pi)-chains 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (8) UsableRulesProof (EQUIVALENT) 8.66/3.56 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (9) 8.66/3.56 Obligation: 8.66/3.56 Pi DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 REV1_IN_GGA(X, .(Y, YS), Z) -> REV1_IN_GGA(Y, YS, Z) 8.66/3.56 8.66/3.56 R is empty. 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 REV1_IN_GGA(x1, x2, x3) = REV1_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 8.66/3.56 We have to consider all (P,R,Pi)-chains 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (10) PiDPToQDPProof (SOUND) 8.66/3.56 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (11) 8.66/3.56 Obligation: 8.66/3.56 Q DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 REV1_IN_GGA(X, .(Y, YS)) -> REV1_IN_GGA(Y, YS) 8.66/3.56 8.66/3.56 R is empty. 8.66/3.56 Q is empty. 8.66/3.56 We have to consider all (P,Q,R)-chains. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (12) QDPSizeChangeProof (EQUIVALENT) 8.66/3.56 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. 8.66/3.56 8.66/3.56 From the DPs we obtained the following set of size-change graphs: 8.66/3.56 *REV1_IN_GGA(X, .(Y, YS)) -> REV1_IN_GGA(Y, YS) 8.66/3.56 The graph contains the following edges 2 > 1, 2 > 2 8.66/3.56 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (13) 8.66/3.56 YES 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (14) 8.66/3.56 Obligation: 8.66/3.56 Pi DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> REV2_IN_GGA(X, XS, YS) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS), ZS) -> U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) -> REV_IN_GA(.(X, VS), ZS) 8.66/3.56 REV_IN_GA(.(X, XS), .(Y, YS)) -> U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> REV_IN_GA(US, VS) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS), ZS) -> REV2_IN_GGA(Y, YS, US) 8.66/3.56 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([], []) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS), .(Y, YS)) -> U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y)) 8.66/3.56 rev1_in_gga(X, [], X) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS), Z) -> U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z)) 8.66/3.56 U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS)) 8.66/3.56 rev2_in_gga(X, [], []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS), ZS) -> U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US)) 8.66/3.56 U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS)) 8.66/3.56 U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS)) 8.66/3.56 U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The argument filtering Pi contains the following mapping: 8.66/3.56 rev_in_ga(x1, x2) = rev_in_ga(x1) 8.66/3.56 8.66/3.56 [] = [] 8.66/3.56 8.66/3.56 rev_out_ga(x1, x2) = rev_out_ga(x1, x2) 8.66/3.56 8.66/3.56 .(x1, x2) = .(x1, x2) 8.66/3.56 8.66/3.56 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) 8.66/3.56 8.66/3.56 rev1_in_gga(x1, x2, x3) = rev1_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev1_out_gga(x1, x2, x3) = rev1_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 rev2_in_gga(x1, x2, x3) = rev2_in_gga(x1, x2) 8.66/3.56 8.66/3.56 rev2_out_gga(x1, x2, x3) = rev2_out_gga(x1, x2, x3) 8.66/3.56 8.66/3.56 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 REV_IN_GA(x1, x2) = REV_IN_GA(x1) 8.66/3.56 8.66/3.56 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) 8.66/3.56 8.66/3.56 REV2_IN_GGA(x1, x2, x3) = REV2_IN_GGA(x1, x2) 8.66/3.56 8.66/3.56 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x5) 8.66/3.56 8.66/3.56 8.66/3.56 We have to consider all (P,R,Pi)-chains 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (15) PiDPToQDPProof (SOUND) 8.66/3.56 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (16) 8.66/3.56 Obligation: 8.66/3.56 Q DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 U1_GA(X, XS, rev1_out_gga(X, XS, Y)) -> REV2_IN_GGA(X, XS) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS)) -> U4_GGA(X, Y, YS, rev2_in_gga(Y, YS)) 8.66/3.56 U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) -> U5_GGA(X, Y, YS, rev_in_ga(US)) 8.66/3.56 U5_GGA(X, Y, YS, rev_out_ga(US, VS)) -> REV_IN_GA(.(X, VS)) 8.66/3.56 REV_IN_GA(.(X, XS)) -> U1_GA(X, XS, rev1_in_gga(X, XS)) 8.66/3.56 U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) -> REV_IN_GA(US) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS)) -> REV2_IN_GGA(Y, YS) 8.66/3.56 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([]) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS)) -> U1_ga(X, XS, rev1_in_gga(X, XS)) 8.66/3.56 rev1_in_gga(X, []) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS)) -> U3_gga(X, Y, YS, rev1_in_gga(Y, YS)) 8.66/3.56 U3_gga(X, Y, YS, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, rev2_in_gga(X, XS)) 8.66/3.56 rev2_in_gga(X, []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS)) -> U4_gga(X, Y, YS, rev2_in_gga(Y, YS)) 8.66/3.56 U4_gga(X, Y, YS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, rev_in_ga(US)) 8.66/3.56 U5_gga(X, Y, YS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, rev_in_ga(.(X, VS))) 8.66/3.56 U6_gga(X, Y, YS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The set Q consists of the following terms: 8.66/3.56 8.66/3.56 rev_in_ga(x0) 8.66/3.56 rev1_in_gga(x0, x1) 8.66/3.56 U3_gga(x0, x1, x2, x3) 8.66/3.56 U1_ga(x0, x1, x2) 8.66/3.56 rev2_in_gga(x0, x1) 8.66/3.56 U4_gga(x0, x1, x2, x3) 8.66/3.56 U5_gga(x0, x1, x2, x3) 8.66/3.56 U6_gga(x0, x1, x2, x3) 8.66/3.56 U2_ga(x0, x1, x2, x3) 8.66/3.56 8.66/3.56 We have to consider all (P,Q,R)-chains. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (17) QDPQMonotonicMRRProof (EQUIVALENT) 8.66/3.56 By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. 8.66/3.56 8.66/3.56 Strictly oriented dependency pairs: 8.66/3.56 8.66/3.56 REV_IN_GA(.(X, XS)) -> U1_GA(X, XS, rev1_in_gga(X, XS)) 8.66/3.56 U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) -> REV_IN_GA(US) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS)) -> REV2_IN_GGA(Y, YS) 8.66/3.56 8.66/3.56 8.66/3.56 Used ordering: Polynomial interpretation [POLO]: 8.66/3.56 8.66/3.56 POL(.(x_1, x_2)) = 2 + 2*x_2 8.66/3.56 POL(REV2_IN_GGA(x_1, x_2)) = 2*x_2 8.66/3.56 POL(REV_IN_GA(x_1)) = 2*x_1 8.66/3.56 POL(U1_GA(x_1, x_2, x_3)) = 2*x_2 8.66/3.56 POL(U1_ga(x_1, x_2, x_3)) = 2 + 2*x_2 + 2*x_3 8.66/3.56 POL(U2_ga(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 8.66/3.56 POL(U3_gga(x_1, x_2, x_3, x_4)) = 2*x_4 8.66/3.56 POL(U4_GGA(x_1, x_2, x_3, x_4)) = 2*x_4 8.66/3.56 POL(U4_gga(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 8.66/3.56 POL(U5_GGA(x_1, x_2, x_3, x_4)) = 2*x_4 8.66/3.56 POL(U5_gga(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 8.66/3.56 POL(U6_gga(x_1, x_2, x_3, x_4)) = x_4 8.66/3.56 POL([]) = 0 8.66/3.56 POL(rev1_in_gga(x_1, x_2)) = 2 + x_2 8.66/3.56 POL(rev1_out_gga(x_1, x_2, x_3)) = 2 + x_2 8.66/3.56 POL(rev2_in_gga(x_1, x_2)) = 2 + 2*x_2 8.66/3.56 POL(rev2_out_gga(x_1, x_2, x_3)) = 2 + 2*x_3 8.66/3.56 POL(rev_in_ga(x_1)) = 2 + 2*x_1 8.66/3.56 POL(rev_out_ga(x_1, x_2)) = 2 + 2*x_2 8.66/3.56 8.66/3.56 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (18) 8.66/3.56 Obligation: 8.66/3.56 Q DP problem: 8.66/3.56 The TRS P consists of the following rules: 8.66/3.56 8.66/3.56 U1_GA(X, XS, rev1_out_gga(X, XS, Y)) -> REV2_IN_GGA(X, XS) 8.66/3.56 REV2_IN_GGA(X, .(Y, YS)) -> U4_GGA(X, Y, YS, rev2_in_gga(Y, YS)) 8.66/3.56 U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) -> U5_GGA(X, Y, YS, rev_in_ga(US)) 8.66/3.56 U5_GGA(X, Y, YS, rev_out_ga(US, VS)) -> REV_IN_GA(.(X, VS)) 8.66/3.56 8.66/3.56 The TRS R consists of the following rules: 8.66/3.56 8.66/3.56 rev_in_ga([]) -> rev_out_ga([], []) 8.66/3.56 rev_in_ga(.(X, XS)) -> U1_ga(X, XS, rev1_in_gga(X, XS)) 8.66/3.56 rev1_in_gga(X, []) -> rev1_out_gga(X, [], X) 8.66/3.56 rev1_in_gga(X, .(Y, YS)) -> U3_gga(X, Y, YS, rev1_in_gga(Y, YS)) 8.66/3.56 U3_gga(X, Y, YS, rev1_out_gga(Y, YS, Z)) -> rev1_out_gga(X, .(Y, YS), Z) 8.66/3.56 U1_ga(X, XS, rev1_out_gga(X, XS, Y)) -> U2_ga(X, XS, Y, rev2_in_gga(X, XS)) 8.66/3.56 rev2_in_gga(X, []) -> rev2_out_gga(X, [], []) 8.66/3.56 rev2_in_gga(X, .(Y, YS)) -> U4_gga(X, Y, YS, rev2_in_gga(Y, YS)) 8.66/3.56 U4_gga(X, Y, YS, rev2_out_gga(Y, YS, US)) -> U5_gga(X, Y, YS, rev_in_ga(US)) 8.66/3.56 U5_gga(X, Y, YS, rev_out_ga(US, VS)) -> U6_gga(X, Y, YS, rev_in_ga(.(X, VS))) 8.66/3.56 U6_gga(X, Y, YS, rev_out_ga(.(X, VS), ZS)) -> rev2_out_gga(X, .(Y, YS), ZS) 8.66/3.56 U2_ga(X, XS, Y, rev2_out_gga(X, XS, YS)) -> rev_out_ga(.(X, XS), .(Y, YS)) 8.66/3.56 8.66/3.56 The set Q consists of the following terms: 8.66/3.56 8.66/3.56 rev_in_ga(x0) 8.66/3.56 rev1_in_gga(x0, x1) 8.66/3.56 U3_gga(x0, x1, x2, x3) 8.66/3.56 U1_ga(x0, x1, x2) 8.66/3.56 rev2_in_gga(x0, x1) 8.66/3.56 U4_gga(x0, x1, x2, x3) 8.66/3.56 U5_gga(x0, x1, x2, x3) 8.66/3.56 U6_gga(x0, x1, x2, x3) 8.66/3.56 U2_ga(x0, x1, x2, x3) 8.66/3.56 8.66/3.56 We have to consider all (P,Q,R)-chains. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (19) DependencyGraphProof (EQUIVALENT) 8.66/3.56 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 8.66/3.56 ---------------------------------------- 8.66/3.56 8.66/3.56 (20) 8.66/3.56 TRUE 8.94/3.59 EOF