5.33/2.25 YES 5.63/2.27 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.63/2.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.63/2.27 5.63/2.27 5.63/2.27 Left Termination of the query pattern 5.63/2.27 5.63/2.27 log2(g,a) 5.63/2.27 5.63/2.27 w.r.t. the given Prolog program could successfully be proven: 5.63/2.27 5.63/2.27 (0) Prolog 5.63/2.27 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.63/2.27 (2) PiTRS 5.63/2.27 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.63/2.27 (4) PiDP 5.63/2.27 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.63/2.27 (6) AND 5.63/2.27 (7) PiDP 5.63/2.27 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.63/2.27 (9) PiDP 5.63/2.27 (10) PiDPToQDPProof [SOUND, 0 ms] 5.63/2.27 (11) QDP 5.63/2.27 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.63/2.27 (13) YES 5.63/2.27 (14) PiDP 5.63/2.27 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.63/2.27 (16) PiDP 5.63/2.27 (17) PiDPToQDPProof [SOUND, 0 ms] 5.63/2.27 (18) QDP 5.63/2.27 (19) MRRProof [EQUIVALENT, 20 ms] 5.63/2.27 (20) QDP 5.63/2.27 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 5.63/2.27 (22) TRUE 5.63/2.27 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (0) 5.63/2.27 Obligation: 5.63/2.27 Clauses: 5.63/2.27 5.63/2.27 log2(X, Y) :- log2(X, 0, Y). 5.63/2.27 log2(0, I, I). 5.63/2.27 log2(s(0), I, I). 5.63/2.27 log2(s(s(X)), I, Y) :- ','(half(s(s(X)), X1), log2(X1, s(I), Y)). 5.63/2.27 half(0, 0). 5.63/2.27 half(s(0), 0). 5.63/2.27 half(s(s(X)), s(Y)) :- half(X, Y). 5.63/2.27 5.63/2.27 5.63/2.27 Query: log2(g,a) 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (1) PrologToPiTRSProof (SOUND) 5.63/2.27 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.63/2.27 5.63/2.27 log2_in_2: (b,f) 5.63/2.27 5.63/2.27 log2_in_3: (b,b,f) 5.63/2.27 5.63/2.27 half_in_2: (b,f) 5.63/2.27 5.63/2.27 Transforming Prolog into the following Term Rewriting System: 5.63/2.27 5.63/2.27 Pi-finite rewrite system: 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 log2_in_gga(0, I, I) -> log2_out_gga(0, I, I) 5.63/2.27 log2_in_gga(s(0), I, I) -> log2_out_gga(s(0), I, I) 5.63/2.27 log2_in_gga(s(s(X)), I, Y) -> U2_gga(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 U2_gga(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_gga(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U3_gga(X, I, Y, log2_out_gga(X1, s(I), Y)) -> log2_out_gga(s(s(X)), I, Y) 5.63/2.27 U1_ga(X, Y, log2_out_gga(X, 0, Y)) -> log2_out_ga(X, Y) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.63/2.27 5.63/2.27 U1_ga(x1, x2, x3) = U1_ga(x3) 5.63/2.27 5.63/2.27 log2_in_gga(x1, x2, x3) = log2_in_gga(x1, x2) 5.63/2.27 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 log2_out_gga(x1, x2, x3) = log2_out_gga(x3) 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 5.63/2.27 5.63/2.27 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.63/2.27 5.63/2.27 5.63/2.27 5.63/2.27 5.63/2.27 5.63/2.27 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.63/2.27 5.63/2.27 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (2) 5.63/2.27 Obligation: 5.63/2.27 Pi-finite rewrite system: 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 log2_in_gga(0, I, I) -> log2_out_gga(0, I, I) 5.63/2.27 log2_in_gga(s(0), I, I) -> log2_out_gga(s(0), I, I) 5.63/2.27 log2_in_gga(s(s(X)), I, Y) -> U2_gga(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 U2_gga(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_gga(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U3_gga(X, I, Y, log2_out_gga(X1, s(I), Y)) -> log2_out_gga(s(s(X)), I, Y) 5.63/2.27 U1_ga(X, Y, log2_out_gga(X, 0, Y)) -> log2_out_ga(X, Y) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.63/2.27 5.63/2.27 U1_ga(x1, x2, x3) = U1_ga(x3) 5.63/2.27 5.63/2.27 log2_in_gga(x1, x2, x3) = log2_in_gga(x1, x2) 5.63/2.27 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 log2_out_gga(x1, x2, x3) = log2_out_gga(x3) 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 5.63/2.27 5.63/2.27 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.63/2.27 5.63/2.27 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (3) DependencyPairsProof (EQUIVALENT) 5.63/2.27 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.63/2.27 Pi DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 LOG2_IN_GA(X, Y) -> U1_GA(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 LOG2_IN_GA(X, Y) -> LOG2_IN_GGA(X, 0, Y) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I, Y) -> U2_GGA(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I, Y) -> HALF_IN_GA(s(s(X)), X1) 5.63/2.27 HALF_IN_GA(s(s(X)), s(Y)) -> U4_GA(X, Y, half_in_ga(X, Y)) 5.63/2.27 HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) 5.63/2.27 U2_GGA(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_GGA(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U2_GGA(X, I, Y, half_out_ga(s(s(X)), X1)) -> LOG2_IN_GGA(X1, s(I), Y) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 log2_in_gga(0, I, I) -> log2_out_gga(0, I, I) 5.63/2.27 log2_in_gga(s(0), I, I) -> log2_out_gga(s(0), I, I) 5.63/2.27 log2_in_gga(s(s(X)), I, Y) -> U2_gga(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 U2_gga(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_gga(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U3_gga(X, I, Y, log2_out_gga(X1, s(I), Y)) -> log2_out_gga(s(s(X)), I, Y) 5.63/2.27 U1_ga(X, Y, log2_out_gga(X, 0, Y)) -> log2_out_ga(X, Y) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.63/2.27 5.63/2.27 U1_ga(x1, x2, x3) = U1_ga(x3) 5.63/2.27 5.63/2.27 log2_in_gga(x1, x2, x3) = log2_in_gga(x1, x2) 5.63/2.27 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 log2_out_gga(x1, x2, x3) = log2_out_gga(x3) 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 5.63/2.27 5.63/2.27 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.63/2.27 5.63/2.27 LOG2_IN_GA(x1, x2) = LOG2_IN_GA(x1) 5.63/2.27 5.63/2.27 U1_GA(x1, x2, x3) = U1_GA(x3) 5.63/2.27 5.63/2.27 LOG2_IN_GGA(x1, x2, x3) = LOG2_IN_GGA(x1, x2) 5.63/2.27 5.63/2.27 U2_GGA(x1, x2, x3, x4) = U2_GGA(x2, x4) 5.63/2.27 5.63/2.27 HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) 5.63/2.27 5.63/2.27 U4_GA(x1, x2, x3) = U4_GA(x3) 5.63/2.27 5.63/2.27 U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) 5.63/2.27 5.63/2.27 5.63/2.27 We have to consider all (P,R,Pi)-chains 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (4) 5.63/2.27 Obligation: 5.63/2.27 Pi DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 LOG2_IN_GA(X, Y) -> U1_GA(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 LOG2_IN_GA(X, Y) -> LOG2_IN_GGA(X, 0, Y) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I, Y) -> U2_GGA(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I, Y) -> HALF_IN_GA(s(s(X)), X1) 5.63/2.27 HALF_IN_GA(s(s(X)), s(Y)) -> U4_GA(X, Y, half_in_ga(X, Y)) 5.63/2.27 HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) 5.63/2.27 U2_GGA(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_GGA(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U2_GGA(X, I, Y, half_out_ga(s(s(X)), X1)) -> LOG2_IN_GGA(X1, s(I), Y) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 log2_in_gga(0, I, I) -> log2_out_gga(0, I, I) 5.63/2.27 log2_in_gga(s(0), I, I) -> log2_out_gga(s(0), I, I) 5.63/2.27 log2_in_gga(s(s(X)), I, Y) -> U2_gga(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 U2_gga(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_gga(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U3_gga(X, I, Y, log2_out_gga(X1, s(I), Y)) -> log2_out_gga(s(s(X)), I, Y) 5.63/2.27 U1_ga(X, Y, log2_out_gga(X, 0, Y)) -> log2_out_ga(X, Y) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.63/2.27 5.63/2.27 U1_ga(x1, x2, x3) = U1_ga(x3) 5.63/2.27 5.63/2.27 log2_in_gga(x1, x2, x3) = log2_in_gga(x1, x2) 5.63/2.27 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 log2_out_gga(x1, x2, x3) = log2_out_gga(x3) 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 5.63/2.27 5.63/2.27 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.63/2.27 5.63/2.27 LOG2_IN_GA(x1, x2) = LOG2_IN_GA(x1) 5.63/2.27 5.63/2.27 U1_GA(x1, x2, x3) = U1_GA(x3) 5.63/2.27 5.63/2.27 LOG2_IN_GGA(x1, x2, x3) = LOG2_IN_GGA(x1, x2) 5.63/2.27 5.63/2.27 U2_GGA(x1, x2, x3, x4) = U2_GGA(x2, x4) 5.63/2.27 5.63/2.27 HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) 5.63/2.27 5.63/2.27 U4_GA(x1, x2, x3) = U4_GA(x3) 5.63/2.27 5.63/2.27 U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) 5.63/2.27 5.63/2.27 5.63/2.27 We have to consider all (P,R,Pi)-chains 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (5) DependencyGraphProof (EQUIVALENT) 5.63/2.27 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (6) 5.63/2.27 Complex Obligation (AND) 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (7) 5.63/2.27 Obligation: 5.63/2.27 Pi DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 log2_in_gga(0, I, I) -> log2_out_gga(0, I, I) 5.63/2.27 log2_in_gga(s(0), I, I) -> log2_out_gga(s(0), I, I) 5.63/2.27 log2_in_gga(s(s(X)), I, Y) -> U2_gga(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 U2_gga(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_gga(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U3_gga(X, I, Y, log2_out_gga(X1, s(I), Y)) -> log2_out_gga(s(s(X)), I, Y) 5.63/2.27 U1_ga(X, Y, log2_out_gga(X, 0, Y)) -> log2_out_ga(X, Y) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.63/2.27 5.63/2.27 U1_ga(x1, x2, x3) = U1_ga(x3) 5.63/2.27 5.63/2.27 log2_in_gga(x1, x2, x3) = log2_in_gga(x1, x2) 5.63/2.27 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 log2_out_gga(x1, x2, x3) = log2_out_gga(x3) 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 5.63/2.27 5.63/2.27 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.63/2.27 5.63/2.27 HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) 5.63/2.27 5.63/2.27 5.63/2.27 We have to consider all (P,R,Pi)-chains 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (8) UsableRulesProof (EQUIVALENT) 5.63/2.27 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (9) 5.63/2.27 Obligation: 5.63/2.27 Pi DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) 5.63/2.27 5.63/2.27 R is empty. 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) 5.63/2.27 5.63/2.27 5.63/2.27 We have to consider all (P,R,Pi)-chains 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (10) PiDPToQDPProof (SOUND) 5.63/2.27 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (11) 5.63/2.27 Obligation: 5.63/2.27 Q DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 HALF_IN_GA(s(s(X))) -> HALF_IN_GA(X) 5.63/2.27 5.63/2.27 R is empty. 5.63/2.27 Q is empty. 5.63/2.27 We have to consider all (P,Q,R)-chains. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (12) QDPSizeChangeProof (EQUIVALENT) 5.63/2.27 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. 5.63/2.27 5.63/2.27 From the DPs we obtained the following set of size-change graphs: 5.63/2.27 *HALF_IN_GA(s(s(X))) -> HALF_IN_GA(X) 5.63/2.27 The graph contains the following edges 1 > 1 5.63/2.27 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (13) 5.63/2.27 YES 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (14) 5.63/2.27 Obligation: 5.63/2.27 Pi DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 U2_GGA(X, I, Y, half_out_ga(s(s(X)), X1)) -> LOG2_IN_GGA(X1, s(I), Y) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I, Y) -> U2_GGA(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 log2_in_ga(X, Y) -> U1_ga(X, Y, log2_in_gga(X, 0, Y)) 5.63/2.27 log2_in_gga(0, I, I) -> log2_out_gga(0, I, I) 5.63/2.27 log2_in_gga(s(0), I, I) -> log2_out_gga(s(0), I, I) 5.63/2.27 log2_in_gga(s(s(X)), I, Y) -> U2_gga(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 U2_gga(X, I, Y, half_out_ga(s(s(X)), X1)) -> U3_gga(X, I, Y, log2_in_gga(X1, s(I), Y)) 5.63/2.27 U3_gga(X, I, Y, log2_out_gga(X1, s(I), Y)) -> log2_out_gga(s(s(X)), I, Y) 5.63/2.27 U1_ga(X, Y, log2_out_gga(X, 0, Y)) -> log2_out_ga(X, Y) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 log2_in_ga(x1, x2) = log2_in_ga(x1) 5.63/2.27 5.63/2.27 U1_ga(x1, x2, x3) = U1_ga(x3) 5.63/2.27 5.63/2.27 log2_in_gga(x1, x2, x3) = log2_in_gga(x1, x2) 5.63/2.27 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 log2_out_gga(x1, x2, x3) = log2_out_gga(x3) 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 5.63/2.27 5.63/2.27 log2_out_ga(x1, x2) = log2_out_ga(x2) 5.63/2.27 5.63/2.27 LOG2_IN_GGA(x1, x2, x3) = LOG2_IN_GGA(x1, x2) 5.63/2.27 5.63/2.27 U2_GGA(x1, x2, x3, x4) = U2_GGA(x2, x4) 5.63/2.27 5.63/2.27 5.63/2.27 We have to consider all (P,R,Pi)-chains 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (15) UsableRulesProof (EQUIVALENT) 5.63/2.27 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (16) 5.63/2.27 Obligation: 5.63/2.27 Pi DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 U2_GGA(X, I, Y, half_out_ga(s(s(X)), X1)) -> LOG2_IN_GGA(X1, s(I), Y) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I, Y) -> U2_GGA(X, I, Y, half_in_ga(s(s(X)), X1)) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 half_in_ga(s(s(X)), s(Y)) -> U4_ga(X, Y, half_in_ga(X, Y)) 5.63/2.27 U4_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) 5.63/2.27 half_in_ga(0, 0) -> half_out_ga(0, 0) 5.63/2.27 half_in_ga(s(0), 0) -> half_out_ga(s(0), 0) 5.63/2.27 5.63/2.27 The argument filtering Pi contains the following mapping: 5.63/2.27 0 = 0 5.63/2.27 5.63/2.27 s(x1) = s(x1) 5.63/2.27 5.63/2.27 half_in_ga(x1, x2) = half_in_ga(x1) 5.63/2.27 5.63/2.27 half_out_ga(x1, x2) = half_out_ga(x2) 5.63/2.27 5.63/2.27 U4_ga(x1, x2, x3) = U4_ga(x3) 5.63/2.27 5.63/2.27 LOG2_IN_GGA(x1, x2, x3) = LOG2_IN_GGA(x1, x2) 5.63/2.27 5.63/2.27 U2_GGA(x1, x2, x3, x4) = U2_GGA(x2, x4) 5.63/2.27 5.63/2.27 5.63/2.27 We have to consider all (P,R,Pi)-chains 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (17) PiDPToQDPProof (SOUND) 5.63/2.27 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (18) 5.63/2.27 Obligation: 5.63/2.27 Q DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 U2_GGA(I, half_out_ga(X1)) -> LOG2_IN_GGA(X1, s(I)) 5.63/2.27 LOG2_IN_GGA(s(s(X)), I) -> U2_GGA(I, half_in_ga(s(s(X)))) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 half_in_ga(s(s(X))) -> U4_ga(half_in_ga(X)) 5.63/2.27 U4_ga(half_out_ga(Y)) -> half_out_ga(s(Y)) 5.63/2.27 half_in_ga(0) -> half_out_ga(0) 5.63/2.27 half_in_ga(s(0)) -> half_out_ga(0) 5.63/2.27 5.63/2.27 The set Q consists of the following terms: 5.63/2.27 5.63/2.27 half_in_ga(x0) 5.63/2.27 U4_ga(x0) 5.63/2.27 5.63/2.27 We have to consider all (P,Q,R)-chains. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (19) MRRProof (EQUIVALENT) 5.63/2.27 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. 5.63/2.27 5.63/2.27 Strictly oriented dependency pairs: 5.63/2.27 5.63/2.27 LOG2_IN_GGA(s(s(X)), I) -> U2_GGA(I, half_in_ga(s(s(X)))) 5.63/2.27 5.63/2.27 Strictly oriented rules of the TRS R: 5.63/2.27 5.63/2.27 half_in_ga(s(0)) -> half_out_ga(0) 5.63/2.27 5.63/2.27 Used ordering: Polynomial interpretation [POLO]: 5.63/2.27 5.63/2.27 POL(0) = 0 5.63/2.27 POL(LOG2_IN_GGA(x_1, x_2)) = 2*x_1 + x_2 5.63/2.27 POL(U2_GGA(x_1, x_2)) = 1 + x_1 + x_2 5.63/2.27 POL(U4_ga(x_1)) = 2 + x_1 5.63/2.27 POL(half_in_ga(x_1)) = x_1 5.63/2.27 POL(half_out_ga(x_1)) = 2*x_1 5.63/2.27 POL(s(x_1)) = 1 + x_1 5.63/2.27 5.63/2.27 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (20) 5.63/2.27 Obligation: 5.63/2.27 Q DP problem: 5.63/2.27 The TRS P consists of the following rules: 5.63/2.27 5.63/2.27 U2_GGA(I, half_out_ga(X1)) -> LOG2_IN_GGA(X1, s(I)) 5.63/2.27 5.63/2.27 The TRS R consists of the following rules: 5.63/2.27 5.63/2.27 half_in_ga(s(s(X))) -> U4_ga(half_in_ga(X)) 5.63/2.27 U4_ga(half_out_ga(Y)) -> half_out_ga(s(Y)) 5.63/2.27 half_in_ga(0) -> half_out_ga(0) 5.63/2.27 5.63/2.27 The set Q consists of the following terms: 5.63/2.27 5.63/2.27 half_in_ga(x0) 5.63/2.27 U4_ga(x0) 5.63/2.27 5.63/2.27 We have to consider all (P,Q,R)-chains. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (21) DependencyGraphProof (EQUIVALENT) 5.63/2.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.63/2.27 ---------------------------------------- 5.63/2.27 5.63/2.27 (22) 5.63/2.27 TRUE 5.63/2.31 EOF