3.64/1.68 YES 3.64/1.69 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 3.64/1.69 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.64/1.69 3.64/1.69 3.64/1.69 Left Termination of the query pattern 3.64/1.69 3.64/1.69 select(g,g,a) 3.64/1.69 3.64/1.69 w.r.t. the given Prolog program could successfully be proven: 3.64/1.69 3.64/1.69 (0) Prolog 3.64/1.69 (1) PrologToPiTRSProof [SOUND, 0 ms] 3.64/1.69 (2) PiTRS 3.64/1.69 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 3.64/1.69 (4) PiDP 3.64/1.69 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 3.64/1.69 (6) PiDP 3.64/1.69 (7) UsableRulesProof [EQUIVALENT, 0 ms] 3.64/1.69 (8) PiDP 3.64/1.69 (9) PiDPToQDPProof [SOUND, 0 ms] 3.64/1.69 (10) QDP 3.64/1.69 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.64/1.69 (12) YES 3.64/1.69 3.64/1.69 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (0) 3.64/1.69 Obligation: 3.64/1.69 Clauses: 3.64/1.69 3.64/1.69 select(X, .(X, Xs), Xs). 3.64/1.69 select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs). 3.64/1.69 3.64/1.69 3.64/1.69 Query: select(g,g,a) 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (1) PrologToPiTRSProof (SOUND) 3.64/1.69 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 3.64/1.69 3.64/1.69 select_in_3: (b,b,f) 3.64/1.69 3.64/1.69 Transforming Prolog into the following Term Rewriting System: 3.64/1.69 3.64/1.69 Pi-finite rewrite system: 3.64/1.69 The TRS R consists of the following rules: 3.64/1.69 3.64/1.69 select_in_gga(X, .(X, Xs), Xs) -> select_out_gga(X, .(X, Xs), Xs) 3.64/1.69 select_in_gga(X, .(Y, Xs), .(Y, Zs)) -> U1_gga(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 U1_gga(X, Y, Xs, Zs, select_out_gga(X, Xs, Zs)) -> select_out_gga(X, .(Y, Xs), .(Y, Zs)) 3.64/1.69 3.64/1.69 The argument filtering Pi contains the following mapping: 3.64/1.69 select_in_gga(x1, x2, x3) = select_in_gga(x1, x2) 3.64/1.69 3.64/1.69 .(x1, x2) = .(x1, x2) 3.64/1.69 3.64/1.69 select_out_gga(x1, x2, x3) = select_out_gga(x1, x2, x3) 3.64/1.69 3.64/1.69 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 3.64/1.69 3.64/1.69 3.64/1.69 3.64/1.69 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 3.64/1.69 3.64/1.69 3.64/1.69 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (2) 3.64/1.69 Obligation: 3.64/1.69 Pi-finite rewrite system: 3.64/1.69 The TRS R consists of the following rules: 3.64/1.69 3.64/1.69 select_in_gga(X, .(X, Xs), Xs) -> select_out_gga(X, .(X, Xs), Xs) 3.64/1.69 select_in_gga(X, .(Y, Xs), .(Y, Zs)) -> U1_gga(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 U1_gga(X, Y, Xs, Zs, select_out_gga(X, Xs, Zs)) -> select_out_gga(X, .(Y, Xs), .(Y, Zs)) 3.64/1.69 3.64/1.69 The argument filtering Pi contains the following mapping: 3.64/1.69 select_in_gga(x1, x2, x3) = select_in_gga(x1, x2) 3.64/1.69 3.64/1.69 .(x1, x2) = .(x1, x2) 3.64/1.69 3.64/1.69 select_out_gga(x1, x2, x3) = select_out_gga(x1, x2, x3) 3.64/1.69 3.64/1.69 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 3.64/1.69 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (3) DependencyPairsProof (EQUIVALENT) 3.64/1.69 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 3.64/1.69 Pi DP problem: 3.64/1.69 The TRS P consists of the following rules: 3.64/1.69 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs), .(Y, Zs)) -> U1_GGA(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GGA(X, Xs, Zs) 3.64/1.69 3.64/1.69 The TRS R consists of the following rules: 3.64/1.69 3.64/1.69 select_in_gga(X, .(X, Xs), Xs) -> select_out_gga(X, .(X, Xs), Xs) 3.64/1.69 select_in_gga(X, .(Y, Xs), .(Y, Zs)) -> U1_gga(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 U1_gga(X, Y, Xs, Zs, select_out_gga(X, Xs, Zs)) -> select_out_gga(X, .(Y, Xs), .(Y, Zs)) 3.64/1.69 3.64/1.69 The argument filtering Pi contains the following mapping: 3.64/1.69 select_in_gga(x1, x2, x3) = select_in_gga(x1, x2) 3.64/1.69 3.64/1.69 .(x1, x2) = .(x1, x2) 3.64/1.69 3.64/1.69 select_out_gga(x1, x2, x3) = select_out_gga(x1, x2, x3) 3.64/1.69 3.64/1.69 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 SELECT_IN_GGA(x1, x2, x3) = SELECT_IN_GGA(x1, x2) 3.64/1.69 3.64/1.69 U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 3.64/1.69 We have to consider all (P,R,Pi)-chains 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (4) 3.64/1.69 Obligation: 3.64/1.69 Pi DP problem: 3.64/1.69 The TRS P consists of the following rules: 3.64/1.69 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs), .(Y, Zs)) -> U1_GGA(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GGA(X, Xs, Zs) 3.64/1.69 3.64/1.69 The TRS R consists of the following rules: 3.64/1.69 3.64/1.69 select_in_gga(X, .(X, Xs), Xs) -> select_out_gga(X, .(X, Xs), Xs) 3.64/1.69 select_in_gga(X, .(Y, Xs), .(Y, Zs)) -> U1_gga(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 U1_gga(X, Y, Xs, Zs, select_out_gga(X, Xs, Zs)) -> select_out_gga(X, .(Y, Xs), .(Y, Zs)) 3.64/1.69 3.64/1.69 The argument filtering Pi contains the following mapping: 3.64/1.69 select_in_gga(x1, x2, x3) = select_in_gga(x1, x2) 3.64/1.69 3.64/1.69 .(x1, x2) = .(x1, x2) 3.64/1.69 3.64/1.69 select_out_gga(x1, x2, x3) = select_out_gga(x1, x2, x3) 3.64/1.69 3.64/1.69 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 SELECT_IN_GGA(x1, x2, x3) = SELECT_IN_GGA(x1, x2) 3.64/1.69 3.64/1.69 U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 3.64/1.69 We have to consider all (P,R,Pi)-chains 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (5) DependencyGraphProof (EQUIVALENT) 3.64/1.69 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (6) 3.64/1.69 Obligation: 3.64/1.69 Pi DP problem: 3.64/1.69 The TRS P consists of the following rules: 3.64/1.69 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GGA(X, Xs, Zs) 3.64/1.69 3.64/1.69 The TRS R consists of the following rules: 3.64/1.69 3.64/1.69 select_in_gga(X, .(X, Xs), Xs) -> select_out_gga(X, .(X, Xs), Xs) 3.64/1.69 select_in_gga(X, .(Y, Xs), .(Y, Zs)) -> U1_gga(X, Y, Xs, Zs, select_in_gga(X, Xs, Zs)) 3.64/1.69 U1_gga(X, Y, Xs, Zs, select_out_gga(X, Xs, Zs)) -> select_out_gga(X, .(Y, Xs), .(Y, Zs)) 3.64/1.69 3.64/1.69 The argument filtering Pi contains the following mapping: 3.64/1.69 select_in_gga(x1, x2, x3) = select_in_gga(x1, x2) 3.64/1.69 3.64/1.69 .(x1, x2) = .(x1, x2) 3.64/1.69 3.64/1.69 select_out_gga(x1, x2, x3) = select_out_gga(x1, x2, x3) 3.64/1.69 3.64/1.69 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5) 3.64/1.69 3.64/1.69 SELECT_IN_GGA(x1, x2, x3) = SELECT_IN_GGA(x1, x2) 3.64/1.69 3.64/1.69 3.64/1.69 We have to consider all (P,R,Pi)-chains 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (7) UsableRulesProof (EQUIVALENT) 3.64/1.69 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (8) 3.64/1.69 Obligation: 3.64/1.69 Pi DP problem: 3.64/1.69 The TRS P consists of the following rules: 3.64/1.69 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_GGA(X, Xs, Zs) 3.64/1.69 3.64/1.69 R is empty. 3.64/1.69 The argument filtering Pi contains the following mapping: 3.64/1.69 .(x1, x2) = .(x1, x2) 3.64/1.69 3.64/1.69 SELECT_IN_GGA(x1, x2, x3) = SELECT_IN_GGA(x1, x2) 3.64/1.69 3.64/1.69 3.64/1.69 We have to consider all (P,R,Pi)-chains 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (9) PiDPToQDPProof (SOUND) 3.64/1.69 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (10) 3.64/1.69 Obligation: 3.64/1.69 Q DP problem: 3.64/1.69 The TRS P consists of the following rules: 3.64/1.69 3.64/1.69 SELECT_IN_GGA(X, .(Y, Xs)) -> SELECT_IN_GGA(X, Xs) 3.64/1.69 3.64/1.69 R is empty. 3.64/1.69 Q is empty. 3.64/1.69 We have to consider all (P,Q,R)-chains. 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (11) QDPSizeChangeProof (EQUIVALENT) 3.64/1.69 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. 3.64/1.69 3.64/1.69 From the DPs we obtained the following set of size-change graphs: 3.64/1.69 *SELECT_IN_GGA(X, .(Y, Xs)) -> SELECT_IN_GGA(X, Xs) 3.64/1.69 The graph contains the following edges 1 >= 1, 2 > 2 3.64/1.69 3.64/1.69 3.64/1.69 ---------------------------------------- 3.64/1.69 3.64/1.69 (12) 3.64/1.69 YES 3.78/1.74 EOF