5.10/2.20 YES 5.40/2.22 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.40/2.22 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.40/2.22 5.40/2.22 5.40/2.22 Left Termination of the query pattern 5.40/2.22 5.40/2.22 perm(g,a) 5.40/2.22 5.40/2.22 w.r.t. the given Prolog program could successfully be proven: 5.40/2.22 5.40/2.22 (0) Prolog 5.40/2.22 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.40/2.22 (2) PiTRS 5.40/2.22 (3) DependencyPairsProof [EQUIVALENT, 3 ms] 5.40/2.22 (4) PiDP 5.40/2.22 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.40/2.22 (6) AND 5.40/2.22 (7) PiDP 5.40/2.22 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.40/2.22 (9) PiDP 5.40/2.22 (10) PiDPToQDPProof [SOUND, 0 ms] 5.40/2.22 (11) QDP 5.40/2.22 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.40/2.22 (13) YES 5.40/2.22 (14) PiDP 5.40/2.22 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.40/2.22 (16) PiDP 5.40/2.22 (17) PiDPToQDPProof [SOUND, 0 ms] 5.40/2.22 (18) QDP 5.40/2.22 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.40/2.22 (20) YES 5.40/2.22 (21) PiDP 5.40/2.22 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.40/2.22 (23) PiDP 5.40/2.22 (24) PiDPToQDPProof [SOUND, 0 ms] 5.40/2.22 (25) QDP 5.40/2.22 (26) MRRProof [EQUIVALENT, 18 ms] 5.40/2.22 (27) QDP 5.40/2.22 (28) DependencyGraphProof [EQUIVALENT, 0 ms] 5.40/2.22 (29) TRUE 5.40/2.22 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (0) 5.40/2.22 Obligation: 5.40/2.22 Clauses: 5.40/2.22 5.40/2.22 app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs). 5.40/2.22 app1([], Ys, Ys). 5.40/2.22 app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs). 5.40/2.22 app2([], Ys, Ys). 5.40/2.22 perm(Xs, .(X, Ys)) :- ','(app2(X1s, .(X, X2s), Xs), ','(app1(X1s, X2s, Zs), perm(Zs, Ys))). 5.40/2.22 perm([], []). 5.40/2.22 5.40/2.22 5.40/2.22 Query: perm(g,a) 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (1) PrologToPiTRSProof (SOUND) 5.40/2.22 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.40/2.22 5.40/2.22 perm_in_2: (b,f) 5.40/2.22 5.40/2.22 app2_in_3: (f,f,b) 5.40/2.22 5.40/2.22 app1_in_3: (b,b,f) 5.40/2.22 5.40/2.22 Transforming Prolog into the following Term Rewriting System: 5.40/2.22 5.40/2.22 Pi-finite rewrite system: 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 5.40/2.22 5.40/2.22 5.40/2.22 5.40/2.22 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.40/2.22 5.40/2.22 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (2) 5.40/2.22 Obligation: 5.40/2.22 Pi-finite rewrite system: 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (3) DependencyPairsProof (EQUIVALENT) 5.40/2.22 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 PERM_IN_GA(Xs, .(X, Ys)) -> APP2_IN_AAG(X1s, .(X, X2s), Xs) 5.40/2.22 APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) 5.40/2.22 U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> APP1_IN_GGA(X1s, X2s, Zs) 5.40/2.22 APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) 5.40/2.22 U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.40/2.22 5.40/2.22 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.40/2.22 5.40/2.22 APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3) 5.40/2.22 5.40/2.22 U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) 5.40/2.22 5.40/2.22 U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) 5.40/2.22 5.40/2.22 APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2) 5.40/2.22 5.40/2.22 U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5) 5.40/2.22 5.40/2.22 U5_GA(x1, x2, x3, x4) = U5_GA(x4) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (4) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 PERM_IN_GA(Xs, .(X, Ys)) -> APP2_IN_AAG(X1s, .(X, X2s), Xs) 5.40/2.22 APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) 5.40/2.22 U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> APP1_IN_GGA(X1s, X2s, Zs) 5.40/2.22 APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) 5.40/2.22 U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.40/2.22 5.40/2.22 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.40/2.22 5.40/2.22 APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3) 5.40/2.22 5.40/2.22 U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) 5.40/2.22 5.40/2.22 U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) 5.40/2.22 5.40/2.22 APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2) 5.40/2.22 5.40/2.22 U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5) 5.40/2.22 5.40/2.22 U5_GA(x1, x2, x3, x4) = U5_GA(x4) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (5) DependencyGraphProof (EQUIVALENT) 5.40/2.22 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (6) 5.40/2.22 Complex Obligation (AND) 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (7) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (8) UsableRulesProof (EQUIVALENT) 5.40/2.22 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (9) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) 5.40/2.22 5.40/2.22 R is empty. 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (10) PiDPToQDPProof (SOUND) 5.40/2.22 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (11) 5.40/2.22 Obligation: 5.40/2.22 Q DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 APP1_IN_GGA(.(Xs), Ys) -> APP1_IN_GGA(Xs, Ys) 5.40/2.22 5.40/2.22 R is empty. 5.40/2.22 Q is empty. 5.40/2.22 We have to consider all (P,Q,R)-chains. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (12) QDPSizeChangeProof (EQUIVALENT) 5.40/2.22 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. 5.40/2.22 5.40/2.22 From the DPs we obtained the following set of size-change graphs: 5.40/2.22 *APP1_IN_GGA(.(Xs), Ys) -> APP1_IN_GGA(Xs, Ys) 5.40/2.22 The graph contains the following edges 1 > 1, 2 >= 2 5.40/2.22 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (13) 5.40/2.22 YES 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (14) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (15) UsableRulesProof (EQUIVALENT) 5.40/2.22 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (16) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) 5.40/2.22 5.40/2.22 R is empty. 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (17) PiDPToQDPProof (SOUND) 5.40/2.22 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (18) 5.40/2.22 Obligation: 5.40/2.22 Q DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 APP2_IN_AAG(.(Zs)) -> APP2_IN_AAG(Zs) 5.40/2.22 5.40/2.22 R is empty. 5.40/2.22 Q is empty. 5.40/2.22 We have to consider all (P,Q,R)-chains. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (19) QDPSizeChangeProof (EQUIVALENT) 5.40/2.22 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. 5.40/2.22 5.40/2.22 From the DPs we obtained the following set of size-change graphs: 5.40/2.22 *APP2_IN_AAG(.(Zs)) -> APP2_IN_AAG(Zs) 5.40/2.22 The graph contains the following edges 1 > 1 5.40/2.22 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (20) 5.40/2.22 YES 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (21) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.40/2.22 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.40/2.22 perm_in_ga([], []) -> perm_out_ga([], []) 5.40/2.22 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.40/2.22 5.40/2.22 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.40/2.22 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 5.40/2.22 5.40/2.22 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.40/2.22 5.40/2.22 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.40/2.22 5.40/2.22 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.40/2.22 5.40/2.22 U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (22) UsableRulesProof (EQUIVALENT) 5.40/2.22 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (23) 5.40/2.22 Obligation: 5.40/2.22 Pi DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) 5.40/2.22 U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.40/2.22 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) 5.40/2.22 app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) 5.40/2.22 app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) 5.40/2.22 app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) 5.40/2.22 U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.40/2.22 5.40/2.22 The argument filtering Pi contains the following mapping: 5.40/2.22 app2_in_aag(x1, x2, x3) = app2_in_aag(x3) 5.40/2.22 5.40/2.22 .(x1, x2) = .(x2) 5.40/2.22 5.40/2.22 U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) 5.40/2.22 5.40/2.22 app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) 5.40/2.22 5.40/2.22 app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) 5.40/2.22 5.40/2.22 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) 5.40/2.22 5.40/2.22 [] = [] 5.40/2.22 5.40/2.22 app1_out_gga(x1, x2, x3) = app1_out_gga(x3) 5.40/2.22 5.40/2.22 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.40/2.22 5.40/2.22 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.40/2.22 5.40/2.22 U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) 5.40/2.22 5.40/2.22 5.40/2.22 We have to consider all (P,R,Pi)-chains 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (24) PiDPToQDPProof (SOUND) 5.40/2.22 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (25) 5.40/2.22 Obligation: 5.40/2.22 Q DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 U3_GA(app2_out_aag(X1s, .(X2s))) -> U4_GA(app1_in_gga(X1s, X2s)) 5.40/2.22 U4_GA(app1_out_gga(Zs)) -> PERM_IN_GA(Zs) 5.40/2.22 PERM_IN_GA(Xs) -> U3_GA(app2_in_aag(Xs)) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 app1_in_gga(.(Xs), Ys) -> U1_gga(app1_in_gga(Xs, Ys)) 5.40/2.22 app1_in_gga([], Ys) -> app1_out_gga(Ys) 5.40/2.22 app2_in_aag(.(Zs)) -> U2_aag(app2_in_aag(Zs)) 5.40/2.22 app2_in_aag(Ys) -> app2_out_aag([], Ys) 5.40/2.22 U1_gga(app1_out_gga(Zs)) -> app1_out_gga(.(Zs)) 5.40/2.22 U2_aag(app2_out_aag(Xs, Ys)) -> app2_out_aag(.(Xs), Ys) 5.40/2.22 5.40/2.22 The set Q consists of the following terms: 5.40/2.22 5.40/2.22 app1_in_gga(x0, x1) 5.40/2.22 app2_in_aag(x0) 5.40/2.22 U1_gga(x0) 5.40/2.22 U2_aag(x0) 5.40/2.22 5.40/2.22 We have to consider all (P,Q,R)-chains. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (26) MRRProof (EQUIVALENT) 5.40/2.22 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. 5.40/2.22 5.40/2.22 Strictly oriented dependency pairs: 5.40/2.22 5.40/2.22 U4_GA(app1_out_gga(Zs)) -> PERM_IN_GA(Zs) 5.40/2.22 PERM_IN_GA(Xs) -> U3_GA(app2_in_aag(Xs)) 5.40/2.22 5.40/2.22 5.40/2.22 Used ordering: Polynomial interpretation [POLO]: 5.40/2.22 5.40/2.22 POL(.(x_1)) = 1 + x_1 5.40/2.22 POL(PERM_IN_GA(x_1)) = 1 + 2*x_1 5.40/2.22 POL(U1_gga(x_1)) = 2 + x_1 5.40/2.22 POL(U2_aag(x_1)) = 1 + x_1 5.40/2.22 POL(U3_GA(x_1)) = 2*x_1 5.40/2.22 POL(U4_GA(x_1)) = 2 + x_1 5.40/2.22 POL([]) = 0 5.40/2.22 POL(app1_in_gga(x_1, x_2)) = 2*x_1 + 2*x_2 5.40/2.22 POL(app1_out_gga(x_1)) = 2*x_1 5.40/2.22 POL(app2_in_aag(x_1)) = x_1 5.40/2.22 POL(app2_out_aag(x_1, x_2)) = x_1 + x_2 5.40/2.22 5.40/2.22 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (27) 5.40/2.22 Obligation: 5.40/2.22 Q DP problem: 5.40/2.22 The TRS P consists of the following rules: 5.40/2.22 5.40/2.22 U3_GA(app2_out_aag(X1s, .(X2s))) -> U4_GA(app1_in_gga(X1s, X2s)) 5.40/2.22 5.40/2.22 The TRS R consists of the following rules: 5.40/2.22 5.40/2.22 app1_in_gga(.(Xs), Ys) -> U1_gga(app1_in_gga(Xs, Ys)) 5.40/2.22 app1_in_gga([], Ys) -> app1_out_gga(Ys) 5.40/2.22 app2_in_aag(.(Zs)) -> U2_aag(app2_in_aag(Zs)) 5.40/2.22 app2_in_aag(Ys) -> app2_out_aag([], Ys) 5.40/2.22 U1_gga(app1_out_gga(Zs)) -> app1_out_gga(.(Zs)) 5.40/2.22 U2_aag(app2_out_aag(Xs, Ys)) -> app2_out_aag(.(Xs), Ys) 5.40/2.22 5.40/2.22 The set Q consists of the following terms: 5.40/2.22 5.40/2.22 app1_in_gga(x0, x1) 5.40/2.22 app2_in_aag(x0) 5.40/2.22 U1_gga(x0) 5.40/2.22 U2_aag(x0) 5.40/2.22 5.40/2.22 We have to consider all (P,Q,R)-chains. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (28) DependencyGraphProof (EQUIVALENT) 5.40/2.22 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.40/2.22 ---------------------------------------- 5.40/2.22 5.40/2.22 (29) 5.40/2.22 TRUE 5.40/2.25 EOF