5.08/2.12 YES 5.08/2.14 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.08/2.14 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.08/2.14 5.08/2.14 5.08/2.14 Left Termination of the query pattern 5.08/2.14 5.08/2.14 perm(g,a) 5.08/2.14 5.08/2.14 w.r.t. the given Prolog program could successfully be proven: 5.08/2.14 5.08/2.14 (0) Prolog 5.08/2.14 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.08/2.14 (2) PiTRS 5.08/2.14 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.08/2.14 (4) PiDP 5.08/2.14 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.08/2.14 (6) AND 5.08/2.14 (7) PiDP 5.08/2.14 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.08/2.14 (9) PiDP 5.08/2.14 (10) PiDPToQDPProof [SOUND, 0 ms] 5.08/2.14 (11) QDP 5.08/2.14 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.08/2.14 (13) YES 5.08/2.14 (14) PiDP 5.08/2.14 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.08/2.14 (16) PiDP 5.08/2.14 (17) PiDPToQDPProof [SOUND, 0 ms] 5.08/2.14 (18) QDP 5.08/2.14 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.08/2.14 (20) YES 5.08/2.14 (21) PiDP 5.08/2.14 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.08/2.14 (23) PiDP 5.08/2.14 (24) PiDPToQDPProof [SOUND, 0 ms] 5.08/2.14 (25) QDP 5.08/2.14 (26) MRRProof [EQUIVALENT, 2 ms] 5.08/2.14 (27) QDP 5.08/2.14 (28) PisEmptyProof [EQUIVALENT, 0 ms] 5.08/2.14 (29) YES 5.08/2.14 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (0) 5.08/2.14 Obligation: 5.08/2.14 Clauses: 5.08/2.14 5.08/2.14 perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))). 5.08/2.14 perm([], []). 5.08/2.14 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 5.08/2.14 app([], Ys, Ys). 5.08/2.14 5.08/2.14 5.08/2.14 Query: perm(g,a) 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (1) PrologToPiTRSProof (SOUND) 5.08/2.14 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.08/2.14 5.08/2.14 perm_in_2: (b,f) 5.08/2.14 5.08/2.14 app_in_3: (f,f,b) (b,b,f) 5.08/2.14 5.08/2.14 Transforming Prolog into the following Term Rewriting System: 5.08/2.14 5.08/2.14 Pi-finite rewrite system: 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 5.08/2.14 5.08/2.14 5.08/2.14 5.08/2.14 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.08/2.14 5.08/2.14 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (2) 5.08/2.14 Obligation: 5.08/2.14 Pi-finite rewrite system: 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (3) DependencyPairsProof (EQUIVALENT) 5.08/2.14 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 PERM_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 PERM_IN_GA(Xs, .(X, Ys)) -> APP_IN_AAG(X1s, .(X, X2s), Xs) 5.08/2.14 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U4_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.08/2.14 U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> APP_IN_GGA(X1s, X2s, Zs) 5.08/2.14 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.08/2.14 U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.08/2.14 5.08/2.14 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.08/2.14 5.08/2.14 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.08/2.14 5.08/2.14 U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x5) 5.08/2.14 5.08/2.14 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.08/2.14 5.08/2.14 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.08/2.14 5.08/2.14 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) 5.08/2.14 5.08/2.14 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (4) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 PERM_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 PERM_IN_GA(Xs, .(X, Ys)) -> APP_IN_AAG(X1s, .(X, X2s), Xs) 5.08/2.14 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U4_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.08/2.14 U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> APP_IN_GGA(X1s, X2s, Zs) 5.08/2.14 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.08/2.14 U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.08/2.14 5.08/2.14 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.08/2.14 5.08/2.14 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.08/2.14 5.08/2.14 U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x5) 5.08/2.14 5.08/2.14 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.08/2.14 5.08/2.14 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.08/2.14 5.08/2.14 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) 5.08/2.14 5.08/2.14 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (5) DependencyGraphProof (EQUIVALENT) 5.08/2.14 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (6) 5.08/2.14 Complex Obligation (AND) 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (7) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (8) UsableRulesProof (EQUIVALENT) 5.08/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (9) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.08/2.14 5.08/2.14 R is empty. 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (10) PiDPToQDPProof (SOUND) 5.08/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (11) 5.08/2.14 Obligation: 5.08/2.14 Q DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys) 5.08/2.14 5.08/2.14 R is empty. 5.08/2.14 Q is empty. 5.08/2.14 We have to consider all (P,Q,R)-chains. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (12) QDPSizeChangeProof (EQUIVALENT) 5.08/2.14 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. 5.08/2.14 5.08/2.14 From the DPs we obtained the following set of size-change graphs: 5.08/2.14 *APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys) 5.08/2.14 The graph contains the following edges 1 > 1, 2 >= 2 5.08/2.14 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (13) 5.08/2.14 YES 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (14) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (15) UsableRulesProof (EQUIVALENT) 5.08/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (16) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.08/2.14 5.08/2.14 R is empty. 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (17) PiDPToQDPProof (SOUND) 5.08/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (18) 5.08/2.14 Obligation: 5.08/2.14 Q DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 APP_IN_AAG(.(Zs)) -> APP_IN_AAG(Zs) 5.08/2.14 5.08/2.14 R is empty. 5.08/2.14 Q is empty. 5.08/2.14 We have to consider all (P,Q,R)-chains. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (19) QDPSizeChangeProof (EQUIVALENT) 5.08/2.14 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. 5.08/2.14 5.08/2.14 From the DPs we obtained the following set of size-change graphs: 5.08/2.14 *APP_IN_AAG(.(Zs)) -> APP_IN_AAG(Zs) 5.08/2.14 The graph contains the following edges 1 > 1 5.08/2.14 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (20) 5.08/2.14 YES 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (21) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.08/2.14 PERM_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 perm_in_ga(Xs, .(X, Ys)) -> U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.08/2.14 perm_in_ga([], []) -> perm_out_ga([], []) 5.08/2.14 U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.08/2.14 5.08/2.14 U1_ga(x1, x2, x3, x4) = U1_ga(x4) 5.08/2.14 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.08/2.14 5.08/2.14 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.08/2.14 5.08/2.14 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.08/2.14 5.08/2.14 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.08/2.14 5.08/2.14 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (22) UsableRulesProof (EQUIVALENT) 5.08/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (23) 5.08/2.14 Obligation: 5.08/2.14 Pi DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.08/2.14 U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.08/2.14 PERM_IN_GA(Xs, .(X, Ys)) -> U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.08/2.14 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 5.08/2.14 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.08/2.14 app_in_aag([], Ys, Ys) -> app_out_aag([], Ys, Ys) 5.08/2.14 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.08/2.14 5.08/2.14 The argument filtering Pi contains the following mapping: 5.08/2.14 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.08/2.14 5.08/2.14 .(x1, x2) = .(x2) 5.08/2.14 5.08/2.14 U4_aag(x1, x2, x3, x4, x5) = U4_aag(x5) 5.08/2.14 5.08/2.14 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.08/2.14 5.08/2.14 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.08/2.14 5.08/2.14 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 5.08/2.14 5.08/2.14 [] = [] 5.08/2.14 5.08/2.14 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.08/2.14 5.08/2.14 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.08/2.14 5.08/2.14 U1_GA(x1, x2, x3, x4) = U1_GA(x4) 5.08/2.14 5.08/2.14 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 5.08/2.14 5.08/2.14 5.08/2.14 We have to consider all (P,R,Pi)-chains 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (24) PiDPToQDPProof (SOUND) 5.08/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (25) 5.08/2.14 Obligation: 5.08/2.14 Q DP problem: 5.08/2.14 The TRS P consists of the following rules: 5.08/2.14 5.08/2.14 U1_GA(app_out_aag(X1s, .(X2s))) -> U2_GA(app_in_gga(X1s, X2s)) 5.08/2.14 U2_GA(app_out_gga(Zs)) -> PERM_IN_GA(Zs) 5.08/2.14 PERM_IN_GA(Xs) -> U1_GA(app_in_aag(Xs)) 5.08/2.14 5.08/2.14 The TRS R consists of the following rules: 5.08/2.14 5.08/2.14 app_in_gga(.(Xs), Ys) -> U4_gga(app_in_gga(Xs, Ys)) 5.08/2.14 app_in_gga([], Ys) -> app_out_gga(Ys) 5.08/2.14 app_in_aag(.(Zs)) -> U4_aag(app_in_aag(Zs)) 5.08/2.14 app_in_aag(Ys) -> app_out_aag([], Ys) 5.08/2.14 U4_gga(app_out_gga(Zs)) -> app_out_gga(.(Zs)) 5.08/2.14 U4_aag(app_out_aag(Xs, Ys)) -> app_out_aag(.(Xs), Ys) 5.08/2.14 5.08/2.14 The set Q consists of the following terms: 5.08/2.14 5.08/2.14 app_in_gga(x0, x1) 5.08/2.14 app_in_aag(x0) 5.08/2.14 U4_gga(x0) 5.08/2.14 U4_aag(x0) 5.08/2.14 5.08/2.14 We have to consider all (P,Q,R)-chains. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (26) MRRProof (EQUIVALENT) 5.08/2.14 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. 5.08/2.14 5.08/2.14 Strictly oriented dependency pairs: 5.08/2.14 5.08/2.14 U1_GA(app_out_aag(X1s, .(X2s))) -> U2_GA(app_in_gga(X1s, X2s)) 5.08/2.14 U2_GA(app_out_gga(Zs)) -> PERM_IN_GA(Zs) 5.08/2.14 PERM_IN_GA(Xs) -> U1_GA(app_in_aag(Xs)) 5.08/2.14 5.08/2.14 Strictly oriented rules of the TRS R: 5.08/2.14 5.08/2.14 app_in_gga(.(Xs), Ys) -> U4_gga(app_in_gga(Xs, Ys)) 5.08/2.14 app_in_gga([], Ys) -> app_out_gga(Ys) 5.08/2.14 app_in_aag(.(Zs)) -> U4_aag(app_in_aag(Zs)) 5.08/2.14 app_in_aag(Ys) -> app_out_aag([], Ys) 5.08/2.14 U4_gga(app_out_gga(Zs)) -> app_out_gga(.(Zs)) 5.08/2.14 U4_aag(app_out_aag(Xs, Ys)) -> app_out_aag(.(Xs), Ys) 5.08/2.14 5.08/2.14 Used ordering: Knuth-Bendix order [KBO] with precedence:U1_GA_1 > app_in_aag_1 > ._1 > app_in_gga_2 > PERM_IN_GA_1 > U4_gga_1 > U4_aag_1 > U2_GA_1 > app_out_aag_2 > app_out_gga_1 > [] 5.08/2.14 5.08/2.14 and weight map: 5.08/2.14 5.08/2.14 []=2 5.08/2.14 ._1=3 5.08/2.14 U4_gga_1=3 5.08/2.14 app_out_gga_1=7 5.08/2.14 app_in_aag_1=4 5.08/2.14 U4_aag_1=3 5.08/2.14 U1_GA_1=2 5.08/2.14 U2_GA_1=1 5.08/2.14 PERM_IN_GA_1=7 5.08/2.14 app_in_gga_2=5 5.08/2.14 app_out_aag_2=1 5.08/2.14 5.08/2.14 The variable weight is 1 5.08/2.14 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (27) 5.08/2.14 Obligation: 5.08/2.14 Q DP problem: 5.08/2.14 P is empty. 5.08/2.14 R is empty. 5.08/2.14 The set Q consists of the following terms: 5.08/2.14 5.08/2.14 app_in_gga(x0, x1) 5.08/2.14 app_in_aag(x0) 5.08/2.14 U4_gga(x0) 5.08/2.14 U4_aag(x0) 5.08/2.14 5.08/2.14 We have to consider all (P,Q,R)-chains. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (28) PisEmptyProof (EQUIVALENT) 5.08/2.14 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.08/2.14 ---------------------------------------- 5.08/2.14 5.08/2.14 (29) 5.08/2.14 YES 5.08/2.18 EOF