4.35/1.91 YES 4.43/1.93 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.43/1.93 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.43/1.93 4.43/1.93 4.43/1.93 Left Termination of the query pattern 4.43/1.93 4.43/1.93 perm1(g,a) 4.43/1.93 4.43/1.93 w.r.t. the given Prolog program could successfully be proven: 4.43/1.93 4.43/1.93 (0) Prolog 4.43/1.93 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.43/1.93 (2) PiTRS 4.43/1.93 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.43/1.93 (4) PiDP 4.43/1.93 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.43/1.93 (6) AND 4.43/1.93 (7) PiDP 4.43/1.93 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.43/1.93 (9) PiDP 4.43/1.93 (10) PiDPToQDPProof [SOUND, 0 ms] 4.43/1.93 (11) QDP 4.43/1.93 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.43/1.93 (13) YES 4.43/1.93 (14) PiDP 4.43/1.93 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.43/1.93 (16) PiDP 4.43/1.93 (17) PiDPToQDPProof [SOUND, 0 ms] 4.43/1.93 (18) QDP 4.43/1.93 (19) MRRProof [EQUIVALENT, 0 ms] 4.43/1.93 (20) QDP 4.43/1.93 (21) PisEmptyProof [EQUIVALENT, 0 ms] 4.43/1.93 (22) YES 4.43/1.93 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (0) 4.43/1.93 Obligation: 4.43/1.93 Clauses: 4.43/1.93 4.43/1.93 perm1([], []). 4.43/1.93 perm1(Xs, .(X, Ys)) :- ','(select(X, Xs, Zs), perm1(Zs, Ys)). 4.43/1.93 select(X, .(X, Xs), Xs). 4.43/1.93 select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs). 4.43/1.93 4.43/1.93 4.43/1.93 Query: perm1(g,a) 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (1) PrologToPiTRSProof (SOUND) 4.43/1.93 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.43/1.93 4.43/1.93 perm1_in_2: (b,f) 4.43/1.93 4.43/1.93 select_in_3: (f,b,f) 4.43/1.93 4.43/1.93 Transforming Prolog into the following Term Rewriting System: 4.43/1.93 4.43/1.93 Pi-finite rewrite system: 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 perm1_in_ga([], []) -> perm1_out_ga([], []) 4.43/1.93 perm1_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) -> perm1_out_ga(Xs, .(X, Ys)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 perm1_in_ga(x1, x2) = perm1_in_ga(x1) 4.43/1.93 4.43/1.93 [] = [] 4.43/1.93 4.43/1.93 perm1_out_ga(x1, x2) = perm1_out_ga(x2) 4.43/1.93 4.43/1.93 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 4.43/1.93 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 4.43/1.93 4.43/1.93 4.43/1.93 4.43/1.93 4.43/1.93 4.43/1.93 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.43/1.93 4.43/1.93 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (2) 4.43/1.93 Obligation: 4.43/1.93 Pi-finite rewrite system: 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 perm1_in_ga([], []) -> perm1_out_ga([], []) 4.43/1.93 perm1_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) -> perm1_out_ga(Xs, .(X, Ys)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 perm1_in_ga(x1, x2) = perm1_in_ga(x1) 4.43/1.93 4.43/1.93 [] = [] 4.43/1.93 4.43/1.93 perm1_out_ga(x1, x2) = perm1_out_ga(x2) 4.43/1.93 4.43/1.93 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 4.43/1.93 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 4.43/1.93 4.43/1.93 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (3) DependencyPairsProof (EQUIVALENT) 4.43/1.93 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.43/1.93 Pi DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 PERM1_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 PERM1_IN_GA(Xs, .(X, Ys)) -> SELECT_IN_AGA(X, Xs, Zs) 4.43/1.93 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> U3_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs) 4.43/1.93 U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_GA(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> PERM1_IN_GA(Zs, Ys) 4.43/1.93 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 perm1_in_ga([], []) -> perm1_out_ga([], []) 4.43/1.93 perm1_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) -> perm1_out_ga(Xs, .(X, Ys)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 perm1_in_ga(x1, x2) = perm1_in_ga(x1) 4.43/1.93 4.43/1.93 [] = [] 4.43/1.93 4.43/1.93 perm1_out_ga(x1, x2) = perm1_out_ga(x2) 4.43/1.93 4.43/1.93 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 4.43/1.93 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 4.43/1.93 4.43/1.93 PERM1_IN_GA(x1, x2) = PERM1_IN_GA(x1) 4.43/1.93 4.43/1.93 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 4.43/1.93 4.43/1.93 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 4.43/1.93 4.43/1.93 U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) 4.43/1.93 4.43/1.93 U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) 4.43/1.93 4.43/1.93 4.43/1.93 We have to consider all (P,R,Pi)-chains 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (4) 4.43/1.93 Obligation: 4.43/1.93 Pi DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 PERM1_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 PERM1_IN_GA(Xs, .(X, Ys)) -> SELECT_IN_AGA(X, Xs, Zs) 4.43/1.93 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> U3_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs) 4.43/1.93 U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_GA(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> PERM1_IN_GA(Zs, Ys) 4.43/1.93 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 perm1_in_ga([], []) -> perm1_out_ga([], []) 4.43/1.93 perm1_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) -> perm1_out_ga(Xs, .(X, Ys)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 perm1_in_ga(x1, x2) = perm1_in_ga(x1) 4.43/1.93 4.43/1.93 [] = [] 4.43/1.93 4.43/1.93 perm1_out_ga(x1, x2) = perm1_out_ga(x2) 4.43/1.93 4.43/1.93 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 4.43/1.93 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 4.43/1.93 4.43/1.93 PERM1_IN_GA(x1, x2) = PERM1_IN_GA(x1) 4.43/1.93 4.43/1.93 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 4.43/1.93 4.43/1.93 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 4.43/1.93 4.43/1.93 U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) 4.43/1.93 4.43/1.93 U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) 4.43/1.93 4.43/1.93 4.43/1.93 We have to consider all (P,R,Pi)-chains 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (5) DependencyGraphProof (EQUIVALENT) 4.43/1.93 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (6) 4.43/1.93 Complex Obligation (AND) 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (7) 4.43/1.93 Obligation: 4.43/1.93 Pi DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs) 4.43/1.93 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 perm1_in_ga([], []) -> perm1_out_ga([], []) 4.43/1.93 perm1_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) -> perm1_out_ga(Xs, .(X, Ys)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 perm1_in_ga(x1, x2) = perm1_in_ga(x1) 4.43/1.93 4.43/1.93 [] = [] 4.43/1.93 4.43/1.93 perm1_out_ga(x1, x2) = perm1_out_ga(x2) 4.43/1.93 4.43/1.93 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 4.43/1.93 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 4.43/1.93 4.43/1.93 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 4.43/1.93 4.43/1.93 4.43/1.93 We have to consider all (P,R,Pi)-chains 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (8) UsableRulesProof (EQUIVALENT) 4.43/1.93 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (9) 4.43/1.93 Obligation: 4.43/1.93 Pi DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs) 4.43/1.93 4.43/1.93 R is empty. 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) 4.43/1.93 4.43/1.93 4.43/1.93 We have to consider all (P,R,Pi)-chains 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (10) PiDPToQDPProof (SOUND) 4.43/1.93 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (11) 4.43/1.93 Obligation: 4.43/1.93 Q DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 SELECT_IN_AGA(.(Y, Xs)) -> SELECT_IN_AGA(Xs) 4.43/1.93 4.43/1.93 R is empty. 4.43/1.93 Q is empty. 4.43/1.93 We have to consider all (P,Q,R)-chains. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (12) QDPSizeChangeProof (EQUIVALENT) 4.43/1.93 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. 4.43/1.93 4.43/1.93 From the DPs we obtained the following set of size-change graphs: 4.43/1.93 *SELECT_IN_AGA(.(Y, Xs)) -> SELECT_IN_AGA(Xs) 4.43/1.93 The graph contains the following edges 1 > 1 4.43/1.93 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (13) 4.43/1.93 YES 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (14) 4.43/1.93 Obligation: 4.43/1.93 Pi DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> PERM1_IN_GA(Zs, Ys) 4.43/1.93 PERM1_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 perm1_in_ga([], []) -> perm1_out_ga([], []) 4.43/1.93 perm1_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys)) 4.43/1.93 U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) -> perm1_out_ga(Xs, .(X, Ys)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 perm1_in_ga(x1, x2) = perm1_in_ga(x1) 4.43/1.93 4.43/1.93 [] = [] 4.43/1.93 4.43/1.93 perm1_out_ga(x1, x2) = perm1_out_ga(x2) 4.43/1.93 4.43/1.93 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 4.43/1.93 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 4.43/1.93 4.43/1.93 PERM1_IN_GA(x1, x2) = PERM1_IN_GA(x1) 4.43/1.93 4.43/1.93 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 4.43/1.93 4.43/1.93 4.43/1.93 We have to consider all (P,R,Pi)-chains 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (15) UsableRulesProof (EQUIVALENT) 4.43/1.93 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (16) 4.43/1.93 Obligation: 4.43/1.93 Pi DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) -> PERM1_IN_GA(Zs, Ys) 4.43/1.93 PERM1_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs)) 4.43/1.93 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) 4.43/1.93 select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs)) 4.43/1.93 U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs)) 4.43/1.93 4.43/1.93 The argument filtering Pi contains the following mapping: 4.43/1.93 select_in_aga(x1, x2, x3) = select_in_aga(x2) 4.43/1.93 4.43/1.93 .(x1, x2) = .(x1, x2) 4.43/1.93 4.43/1.93 select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) 4.43/1.93 4.43/1.93 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.43/1.93 4.43/1.93 PERM1_IN_GA(x1, x2) = PERM1_IN_GA(x1) 4.43/1.93 4.43/1.93 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 4.43/1.93 4.43/1.93 4.43/1.93 We have to consider all (P,R,Pi)-chains 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (17) PiDPToQDPProof (SOUND) 4.43/1.93 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (18) 4.43/1.93 Obligation: 4.43/1.93 Q DP problem: 4.43/1.93 The TRS P consists of the following rules: 4.43/1.93 4.43/1.93 U1_GA(select_out_aga(X, Zs)) -> PERM1_IN_GA(Zs) 4.43/1.93 PERM1_IN_GA(Xs) -> U1_GA(select_in_aga(Xs)) 4.43/1.93 4.43/1.93 The TRS R consists of the following rules: 4.43/1.93 4.43/1.93 select_in_aga(.(X, Xs)) -> select_out_aga(X, Xs) 4.43/1.93 select_in_aga(.(Y, Xs)) -> U3_aga(Y, select_in_aga(Xs)) 4.43/1.93 U3_aga(Y, select_out_aga(X, Zs)) -> select_out_aga(X, .(Y, Zs)) 4.43/1.93 4.43/1.93 The set Q consists of the following terms: 4.43/1.93 4.43/1.93 select_in_aga(x0) 4.43/1.93 U3_aga(x0, x1) 4.43/1.93 4.43/1.93 We have to consider all (P,Q,R)-chains. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (19) MRRProof (EQUIVALENT) 4.43/1.93 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. 4.43/1.93 4.43/1.93 Strictly oriented dependency pairs: 4.43/1.93 4.43/1.93 U1_GA(select_out_aga(X, Zs)) -> PERM1_IN_GA(Zs) 4.43/1.93 PERM1_IN_GA(Xs) -> U1_GA(select_in_aga(Xs)) 4.43/1.93 4.43/1.93 Strictly oriented rules of the TRS R: 4.43/1.93 4.43/1.93 select_in_aga(.(X, Xs)) -> select_out_aga(X, Xs) 4.43/1.93 select_in_aga(.(Y, Xs)) -> U3_aga(Y, select_in_aga(Xs)) 4.43/1.93 U3_aga(Y, select_out_aga(X, Zs)) -> select_out_aga(X, .(Y, Zs)) 4.43/1.93 4.43/1.93 Used ordering: Knuth-Bendix order [KBO] with precedence:U1_GA_1 > ._2 > select_in_aga_1 > U3_aga_2 > PERM1_IN_GA_1 > select_out_aga_2 4.43/1.93 4.43/1.93 and weight map: 4.43/1.93 4.43/1.93 select_in_aga_1=1 4.43/1.93 U1_GA_1=1 4.43/1.93 PERM1_IN_GA_1=3 4.43/1.93 ._2=0 4.43/1.93 select_out_aga_2=1 4.43/1.93 U3_aga_2=0 4.43/1.93 4.43/1.93 The variable weight is 2 4.43/1.93 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (20) 4.43/1.93 Obligation: 4.43/1.93 Q DP problem: 4.43/1.93 P is empty. 4.43/1.93 R is empty. 4.43/1.93 The set Q consists of the following terms: 4.43/1.93 4.43/1.93 select_in_aga(x0) 4.43/1.93 U3_aga(x0, x1) 4.43/1.93 4.43/1.93 We have to consider all (P,Q,R)-chains. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (21) PisEmptyProof (EQUIVALENT) 4.43/1.93 The TRS P is empty. Hence, there is no (P,Q,R) chain. 4.43/1.93 ---------------------------------------- 4.43/1.93 4.43/1.93 (22) 4.43/1.93 YES 4.43/1.96 EOF