5.74/2.46 YES 5.74/2.48 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.74/2.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.74/2.48 5.74/2.48 5.74/2.48 Left Termination of the query pattern 5.74/2.48 5.74/2.48 ss(g,a) 5.74/2.48 5.74/2.48 w.r.t. the given Prolog program could successfully be proven: 5.74/2.48 5.74/2.48 (0) Prolog 5.74/2.48 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.74/2.48 (2) PiTRS 5.74/2.48 (3) DependencyPairsProof [EQUIVALENT, 24 ms] 5.74/2.48 (4) PiDP 5.74/2.48 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.74/2.48 (6) AND 5.74/2.48 (7) PiDP 5.74/2.48 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.74/2.48 (9) PiDP 5.74/2.48 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.74/2.48 (11) QDP 5.74/2.48 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.74/2.48 (13) YES 5.74/2.48 (14) PiDP 5.74/2.48 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.74/2.48 (16) PiDP 5.74/2.48 (17) PiDPToQDPProof [SOUND, 0 ms] 5.74/2.48 (18) QDP 5.74/2.48 (19) UsableRulesReductionPairsProof [EQUIVALENT, 33 ms] 5.74/2.48 (20) QDP 5.74/2.48 (21) MRRProof [EQUIVALENT, 15 ms] 5.74/2.48 (22) QDP 5.74/2.48 (23) PisEmptyProof [EQUIVALENT, 0 ms] 5.74/2.48 (24) YES 5.74/2.48 (25) PiDP 5.74/2.48 (26) UsableRulesProof [EQUIVALENT, 0 ms] 5.74/2.48 (27) PiDP 5.74/2.48 (28) PiDPToQDPProof [SOUND, 0 ms] 5.74/2.48 (29) QDP 5.74/2.48 (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.74/2.48 (31) YES 5.74/2.48 (32) PiDP 5.74/2.48 (33) UsableRulesProof [EQUIVALENT, 0 ms] 5.74/2.48 (34) PiDP 5.74/2.48 (35) PiDPToQDPProof [SOUND, 0 ms] 5.74/2.48 (36) QDP 5.74/2.48 (37) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.74/2.48 (38) YES 5.74/2.48 (39) PiDP 5.74/2.48 (40) UsableRulesProof [EQUIVALENT, 0 ms] 5.74/2.48 (41) PiDP 5.74/2.48 (42) PiDPToQDPProof [SOUND, 0 ms] 5.74/2.48 (43) QDP 5.74/2.48 (44) MRRProof [EQUIVALENT, 0 ms] 5.74/2.48 (45) QDP 5.74/2.48 (46) PisEmptyProof [EQUIVALENT, 0 ms] 5.74/2.48 (47) YES 5.74/2.48 5.74/2.48 5.74/2.48 ---------------------------------------- 5.74/2.48 5.74/2.48 (0) 5.74/2.48 Obligation: 5.74/2.48 Clauses: 5.74/2.48 5.74/2.48 ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)). 5.74/2.48 perm([], []). 5.74/2.48 perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))). 5.74/2.48 app([], X, X). 5.74/2.48 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 5.74/2.48 ordered([]). 5.74/2.48 ordered(.(X1, [])). 5.74/2.48 ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))). 5.74/2.48 less(0, s(X2)). 5.74/2.48 less(s(X), s(Y)) :- less(X, Y). 5.74/2.48 5.74/2.48 5.74/2.48 Query: ss(g,a) 5.74/2.48 ---------------------------------------- 5.74/2.48 5.74/2.48 (1) PrologToPiTRSProof (SOUND) 5.74/2.48 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.74/2.48 5.74/2.48 ss_in_2: (b,f) 5.74/2.48 5.74/2.48 perm_in_2: (b,f) 5.74/2.48 5.74/2.48 app_in_3: (f,f,b) (b,b,f) 5.74/2.48 5.74/2.48 ordered_in_1: (b) 5.74/2.48 5.74/2.48 less_in_2: (b,b) 5.74/2.48 5.74/2.48 Transforming Prolog into the following Term Rewriting System: 5.74/2.48 5.74/2.48 Pi-finite rewrite system: 5.74/2.48 The TRS R consists of the following rules: 5.74/2.48 5.74/2.48 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.48 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.48 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.48 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.48 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.48 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.48 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.48 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.48 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.48 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.48 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.48 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.48 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.48 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.48 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.48 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.48 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.48 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.48 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.48 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.48 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.48 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.48 5.74/2.48 The argument filtering Pi contains the following mapping: 5.74/2.48 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.48 5.74/2.48 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.48 5.74/2.48 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.48 5.74/2.48 [] = [] 5.74/2.48 5.74/2.48 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.48 5.74/2.48 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.48 5.74/2.48 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.48 5.74/2.48 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.48 5.74/2.48 .(x1, x2) = .(x1, x2) 5.74/2.48 5.74/2.48 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.48 5.74/2.48 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.48 5.74/2.48 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.48 5.74/2.48 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.48 5.74/2.48 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.48 5.74/2.48 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.48 5.74/2.48 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.48 5.74/2.48 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.48 5.74/2.48 ordered_out_g(x1) = ordered_out_g 5.74/2.48 5.74/2.48 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.48 5.74/2.48 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.48 5.74/2.48 0 = 0 5.74/2.48 5.74/2.48 s(x1) = s(x1) 5.74/2.48 5.74/2.48 less_out_gg(x1, x2) = less_out_gg 5.74/2.48 5.74/2.48 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.48 5.74/2.48 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.48 5.74/2.48 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.48 5.74/2.48 5.74/2.48 5.74/2.48 5.74/2.48 5.74/2.48 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.74/2.48 5.74/2.48 5.74/2.48 5.74/2.48 ---------------------------------------- 5.74/2.48 5.74/2.48 (2) 5.74/2.48 Obligation: 5.74/2.48 Pi-finite rewrite system: 5.74/2.48 The TRS R consists of the following rules: 5.74/2.48 5.74/2.48 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.48 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.48 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.48 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.48 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.48 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.48 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.48 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.48 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.48 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.48 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.48 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.48 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.48 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.48 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.48 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.48 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.48 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.48 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.48 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.48 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.48 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.48 5.74/2.48 The argument filtering Pi contains the following mapping: 5.74/2.48 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.48 5.74/2.48 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.48 5.74/2.48 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.48 5.74/2.48 [] = [] 5.74/2.48 5.74/2.48 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.48 5.74/2.48 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.48 5.74/2.48 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.48 5.74/2.48 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.48 5.74/2.48 .(x1, x2) = .(x1, x2) 5.74/2.48 5.74/2.48 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (3) DependencyPairsProof (EQUIVALENT) 5.74/2.49 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 SS_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 SS_IN_GA(Xs, Ys) -> PERM_IN_GA(Xs, Ys) 5.74/2.49 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 PERM_IN_GA(Xs, .(X, Ys)) -> APP_IN_AAG(X1s, .(X, X2s), Xs) 5.74/2.49 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U6_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.74/2.49 U3_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 U3_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> APP_IN_GGA(X1s, X2s, Zs) 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U6_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.74/2.49 U4_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U4_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.74/2.49 U1_GA(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_GA(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 U1_GA(Xs, Ys, perm_out_ga(Xs, Ys)) -> ORDERED_IN_G(Ys) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y)) 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y)) 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.74/2.49 U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 SS_IN_GA(x1, x2) = SS_IN_GA(x1) 5.74/2.49 5.74/2.49 U1_GA(x1, x2, x3) = U1_GA(x3) 5.74/2.49 5.74/2.49 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.74/2.49 5.74/2.49 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.74/2.49 5.74/2.49 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.74/2.49 5.74/2.49 U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x1, x5) 5.74/2.49 5.74/2.49 U4_GA(x1, x2, x3, x4) = U4_GA(x2, x4) 5.74/2.49 5.74/2.49 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.74/2.49 5.74/2.49 U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x5) 5.74/2.49 5.74/2.49 U5_GA(x1, x2, x3, x4) = U5_GA(x2, x4) 5.74/2.49 5.74/2.49 U2_GA(x1, x2, x3) = U2_GA(x2, x3) 5.74/2.49 5.74/2.49 ORDERED_IN_G(x1) = ORDERED_IN_G(x1) 5.74/2.49 5.74/2.49 U7_G(x1, x2, x3, x4) = U7_G(x2, x3, x4) 5.74/2.49 5.74/2.49 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.74/2.49 5.74/2.49 U9_GG(x1, x2, x3) = U9_GG(x3) 5.74/2.49 5.74/2.49 U8_G(x1, x2, x3, x4) = U8_G(x4) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (4) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 SS_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 SS_IN_GA(Xs, Ys) -> PERM_IN_GA(Xs, Ys) 5.74/2.49 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 PERM_IN_GA(Xs, .(X, Ys)) -> APP_IN_AAG(X1s, .(X, X2s), Xs) 5.74/2.49 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U6_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.74/2.49 U3_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 U3_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> APP_IN_GGA(X1s, X2s, Zs) 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U6_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.74/2.49 U4_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U4_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.74/2.49 U1_GA(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_GA(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 U1_GA(Xs, Ys, perm_out_ga(Xs, Ys)) -> ORDERED_IN_G(Ys) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y)) 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y)) 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.74/2.49 U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 SS_IN_GA(x1, x2) = SS_IN_GA(x1) 5.74/2.49 5.74/2.49 U1_GA(x1, x2, x3) = U1_GA(x3) 5.74/2.49 5.74/2.49 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.74/2.49 5.74/2.49 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.74/2.49 5.74/2.49 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.74/2.49 5.74/2.49 U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x1, x5) 5.74/2.49 5.74/2.49 U4_GA(x1, x2, x3, x4) = U4_GA(x2, x4) 5.74/2.49 5.74/2.49 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.74/2.49 5.74/2.49 U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x5) 5.74/2.49 5.74/2.49 U5_GA(x1, x2, x3, x4) = U5_GA(x2, x4) 5.74/2.49 5.74/2.49 U2_GA(x1, x2, x3) = U2_GA(x2, x3) 5.74/2.49 5.74/2.49 ORDERED_IN_G(x1) = ORDERED_IN_G(x1) 5.74/2.49 5.74/2.49 U7_G(x1, x2, x3, x4) = U7_G(x2, x3, x4) 5.74/2.49 5.74/2.49 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.74/2.49 5.74/2.49 U9_GG(x1, x2, x3) = U9_GG(x3) 5.74/2.49 5.74/2.49 U8_G(x1, x2, x3, x4) = U8_G(x4) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (5) DependencyGraphProof (EQUIVALENT) 5.74/2.49 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 12 less nodes. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (6) 5.74/2.49 Complex Obligation (AND) 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (7) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (8) UsableRulesProof (EQUIVALENT) 5.74/2.49 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (9) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.74/2.49 5.74/2.49 R is empty. 5.74/2.49 Pi is empty. 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (10) PiDPToQDPProof (EQUIVALENT) 5.74/2.49 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (11) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.74/2.49 5.74/2.49 R is empty. 5.74/2.49 Q is empty. 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (12) QDPSizeChangeProof (EQUIVALENT) 5.74/2.49 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. 5.74/2.49 5.74/2.49 From the DPs we obtained the following set of size-change graphs: 5.74/2.49 *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.74/2.49 The graph contains the following edges 1 > 1, 2 > 2 5.74/2.49 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (13) 5.74/2.49 YES 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (14) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 ORDERED_IN_G(x1) = ORDERED_IN_G(x1) 5.74/2.49 5.74/2.49 U7_G(x1, x2, x3, x4) = U7_G(x2, x3, x4) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (15) UsableRulesProof (EQUIVALENT) 5.74/2.49 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (16) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 ORDERED_IN_G(x1) = ORDERED_IN_G(x1) 5.74/2.49 5.74/2.49 U7_G(x1, x2, x3, x4) = U7_G(x2, x3, x4) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (17) PiDPToQDPProof (SOUND) 5.74/2.49 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (18) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U7_G(Y, Xs, less_out_gg) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 5.74/2.49 U9_gg(less_out_gg) -> less_out_gg 5.74/2.49 5.74/2.49 The set Q consists of the following terms: 5.74/2.49 5.74/2.49 less_in_gg(x0, x1) 5.74/2.49 U9_gg(x0) 5.74/2.49 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (19) UsableRulesReductionPairsProof (EQUIVALENT) 5.74/2.49 By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. 5.74/2.49 5.74/2.49 No dependency pairs are removed. 5.74/2.49 5.74/2.49 The following rules are removed from R: 5.74/2.49 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg 5.74/2.49 Used ordering: POLO with Polynomial interpretation [POLO]: 5.74/2.49 5.74/2.49 POL(.(x_1, x_2)) = 2*x_1 + 2*x_2 5.74/2.49 POL(0) = 0 5.74/2.49 POL(ORDERED_IN_G(x_1)) = x_1 5.74/2.49 POL(U7_G(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 5.74/2.49 POL(U9_gg(x_1)) = x_1 5.74/2.49 POL(less_in_gg(x_1, x_2)) = x_1 + x_2 5.74/2.49 POL(less_out_gg) = 0 5.74/2.49 POL(s(x_1)) = x_1 5.74/2.49 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (20) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U7_G(Y, Xs, less_out_gg) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 5.74/2.49 U9_gg(less_out_gg) -> less_out_gg 5.74/2.49 5.74/2.49 The set Q consists of the following terms: 5.74/2.49 5.74/2.49 less_in_gg(x0, x1) 5.74/2.49 U9_gg(x0) 5.74/2.49 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (21) MRRProof (EQUIVALENT) 5.74/2.49 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. 5.74/2.49 5.74/2.49 Strictly oriented dependency pairs: 5.74/2.49 5.74/2.49 U7_G(Y, Xs, less_out_gg) -> ORDERED_IN_G(.(Y, Xs)) 5.74/2.49 ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 5.74/2.49 5.74/2.49 Used ordering: Polynomial interpretation [POLO]: 5.74/2.49 5.74/2.49 POL(.(x_1, x_2)) = 2*x_1 + 2*x_2 5.74/2.49 POL(ORDERED_IN_G(x_1)) = 2 + x_1 5.74/2.49 POL(U7_G(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 5.74/2.49 POL(U9_gg(x_1)) = x_1 5.74/2.49 POL(less_in_gg(x_1, x_2)) = x_1 + x_2 5.74/2.49 POL(less_out_gg) = 2 5.74/2.49 POL(s(x_1)) = x_1 5.74/2.49 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (22) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 P is empty. 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 5.74/2.49 U9_gg(less_out_gg) -> less_out_gg 5.74/2.49 5.74/2.49 The set Q consists of the following terms: 5.74/2.49 5.74/2.49 less_in_gg(x0, x1) 5.74/2.49 U9_gg(x0) 5.74/2.49 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (23) PisEmptyProof (EQUIVALENT) 5.74/2.49 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (24) 5.74/2.49 YES 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (25) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (26) UsableRulesProof (EQUIVALENT) 5.74/2.49 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (27) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 5.74/2.49 5.74/2.49 R is empty. 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (28) PiDPToQDPProof (SOUND) 5.74/2.49 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (29) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 5.74/2.49 5.74/2.49 R is empty. 5.74/2.49 Q is empty. 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (30) QDPSizeChangeProof (EQUIVALENT) 5.74/2.49 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. 5.74/2.49 5.74/2.49 From the DPs we obtained the following set of size-change graphs: 5.74/2.49 *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 5.74/2.49 The graph contains the following edges 1 > 1, 2 >= 2 5.74/2.49 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (31) 5.74/2.49 YES 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (32) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (33) UsableRulesProof (EQUIVALENT) 5.74/2.49 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (34) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 5.74/2.49 5.74/2.49 R is empty. 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (35) PiDPToQDPProof (SOUND) 5.74/2.49 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (36) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) 5.74/2.49 5.74/2.49 R is empty. 5.74/2.49 Q is empty. 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (37) QDPSizeChangeProof (EQUIVALENT) 5.74/2.49 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. 5.74/2.49 5.74/2.49 From the DPs we obtained the following set of size-change graphs: 5.74/2.49 *APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) 5.74/2.49 The graph contains the following edges 1 > 1 5.74/2.49 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (38) 5.74/2.49 YES 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (39) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U3_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 U4_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.74/2.49 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 ss_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, perm_in_ga(Xs, Ys)) 5.74/2.49 perm_in_ga([], []) -> perm_out_ga([], []) 5.74/2.49 perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U3_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U4_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) 5.74/2.49 U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) 5.74/2.49 U1_ga(Xs, Ys, perm_out_ga(Xs, Ys)) -> U2_ga(Xs, Ys, ordered_in_g(Ys)) 5.74/2.49 ordered_in_g([]) -> ordered_out_g([]) 5.74/2.49 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.74/2.49 ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.74/2.49 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.74/2.49 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 5.74/2.49 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.74/2.49 U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.74/2.49 U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.74/2.49 U2_ga(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ga(Xs, Ys) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 ss_in_ga(x1, x2) = ss_in_ga(x1) 5.74/2.49 5.74/2.49 U1_ga(x1, x2, x3) = U1_ga(x3) 5.74/2.49 5.74/2.49 perm_in_ga(x1, x2) = perm_in_ga(x1) 5.74/2.49 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 perm_out_ga(x1, x2) = perm_out_ga(x2) 5.74/2.49 5.74/2.49 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) 5.74/2.49 5.74/2.49 U2_ga(x1, x2, x3) = U2_ga(x2, x3) 5.74/2.49 5.74/2.49 ordered_in_g(x1) = ordered_in_g(x1) 5.74/2.49 5.74/2.49 ordered_out_g(x1) = ordered_out_g 5.74/2.49 5.74/2.49 U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4) 5.74/2.49 5.74/2.49 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.74/2.49 5.74/2.49 0 = 0 5.74/2.49 5.74/2.49 s(x1) = s(x1) 5.74/2.49 5.74/2.49 less_out_gg(x1, x2) = less_out_gg 5.74/2.49 5.74/2.49 U9_gg(x1, x2, x3) = U9_gg(x3) 5.74/2.49 5.74/2.49 U8_g(x1, x2, x3, x4) = U8_g(x4) 5.74/2.49 5.74/2.49 ss_out_ga(x1, x2) = ss_out_ga(x2) 5.74/2.49 5.74/2.49 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.74/2.49 5.74/2.49 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.74/2.49 5.74/2.49 U4_GA(x1, x2, x3, x4) = U4_GA(x2, x4) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (40) UsableRulesProof (EQUIVALENT) 5.74/2.49 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (41) 5.74/2.49 Obligation: 5.74/2.49 Pi DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U3_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs)) 5.74/2.49 U4_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) 5.74/2.49 PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs)) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 app_in_gga([], X, X) -> app_out_gga([], X, X) 5.74/2.49 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 5.74/2.49 app_in_aag([], X, X) -> app_out_aag([], X, X) 5.74/2.49 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 5.74/2.49 U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 5.74/2.49 5.74/2.49 The argument filtering Pi contains the following mapping: 5.74/2.49 [] = [] 5.74/2.49 5.74/2.49 app_in_aag(x1, x2, x3) = app_in_aag(x3) 5.74/2.49 5.74/2.49 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 5.74/2.49 5.74/2.49 .(x1, x2) = .(x1, x2) 5.74/2.49 5.74/2.49 U6_aag(x1, x2, x3, x4, x5) = U6_aag(x1, x5) 5.74/2.49 5.74/2.49 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 5.74/2.49 5.74/2.49 app_out_gga(x1, x2, x3) = app_out_gga(x3) 5.74/2.49 5.74/2.49 U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) 5.74/2.49 5.74/2.49 PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) 5.74/2.49 5.74/2.49 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 5.74/2.49 5.74/2.49 U4_GA(x1, x2, x3, x4) = U4_GA(x2, x4) 5.74/2.49 5.74/2.49 5.74/2.49 We have to consider all (P,R,Pi)-chains 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (42) PiDPToQDPProof (SOUND) 5.74/2.49 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (43) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 The TRS P consists of the following rules: 5.74/2.49 5.74/2.49 U3_GA(app_out_aag(X1s, .(X, X2s))) -> U4_GA(X, app_in_gga(X1s, X2s)) 5.74/2.49 U4_GA(X, app_out_gga(Zs)) -> PERM_IN_GA(Zs) 5.74/2.49 PERM_IN_GA(Xs) -> U3_GA(app_in_aag(Xs)) 5.74/2.49 5.74/2.49 The TRS R consists of the following rules: 5.74/2.49 5.74/2.49 app_in_gga([], X) -> app_out_gga(X) 5.74/2.49 app_in_gga(.(X, Xs), Ys) -> U6_gga(X, app_in_gga(Xs, Ys)) 5.74/2.49 app_in_aag(X) -> app_out_aag([], X) 5.74/2.49 app_in_aag(.(X, Zs)) -> U6_aag(X, app_in_aag(Zs)) 5.74/2.49 U6_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 5.74/2.49 U6_aag(X, app_out_aag(Xs, Ys)) -> app_out_aag(.(X, Xs), Ys) 5.74/2.49 5.74/2.49 The set Q consists of the following terms: 5.74/2.49 5.74/2.49 app_in_gga(x0, x1) 5.74/2.49 app_in_aag(x0) 5.74/2.49 U6_gga(x0, x1) 5.74/2.49 U6_aag(x0, x1) 5.74/2.49 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (44) MRRProof (EQUIVALENT) 5.74/2.49 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. 5.74/2.49 5.74/2.49 Strictly oriented dependency pairs: 5.74/2.49 5.74/2.49 U3_GA(app_out_aag(X1s, .(X, X2s))) -> U4_GA(X, app_in_gga(X1s, X2s)) 5.74/2.49 U4_GA(X, app_out_gga(Zs)) -> PERM_IN_GA(Zs) 5.74/2.49 PERM_IN_GA(Xs) -> U3_GA(app_in_aag(Xs)) 5.74/2.49 5.74/2.49 Strictly oriented rules of the TRS R: 5.74/2.49 5.74/2.49 app_in_gga([], X) -> app_out_gga(X) 5.74/2.49 app_in_gga(.(X, Xs), Ys) -> U6_gga(X, app_in_gga(Xs, Ys)) 5.74/2.49 app_in_aag(X) -> app_out_aag([], X) 5.74/2.49 app_in_aag(.(X, Zs)) -> U6_aag(X, app_in_aag(Zs)) 5.74/2.49 U6_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 5.74/2.49 U6_aag(X, app_out_aag(Xs, Ys)) -> app_out_aag(.(X, Xs), Ys) 5.74/2.49 5.74/2.49 Used ordering: Knuth-Bendix order [KBO] with precedence:PERM_IN_GA_1 > U4_GA_2 > app_in_aag_1 > ._2 > app_in_gga_2 > [] > U3_GA_1 > U6_aag_2 > app_out_aag_2 > U6_gga_2 > app_out_gga_1 5.74/2.49 5.74/2.49 and weight map: 5.74/2.49 5.74/2.49 []=3 5.74/2.49 app_out_gga_1=3 5.74/2.49 app_in_aag_1=3 5.74/2.49 U3_GA_1=1 5.74/2.49 PERM_IN_GA_1=4 5.74/2.49 app_in_gga_2=0 5.74/2.49 ._2=1 5.74/2.49 U6_gga_2=1 5.74/2.49 app_out_aag_2=0 5.74/2.49 U6_aag_2=1 5.74/2.49 U4_GA_2=1 5.74/2.49 5.74/2.49 The variable weight is 1 5.74/2.49 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (45) 5.74/2.49 Obligation: 5.74/2.49 Q DP problem: 5.74/2.49 P is empty. 5.74/2.49 R is empty. 5.74/2.49 The set Q consists of the following terms: 5.74/2.49 5.74/2.49 app_in_gga(x0, x1) 5.74/2.49 app_in_aag(x0) 5.74/2.49 U6_gga(x0, x1) 5.74/2.49 U6_aag(x0, x1) 5.74/2.49 5.74/2.49 We have to consider all (P,Q,R)-chains. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (46) PisEmptyProof (EQUIVALENT) 5.74/2.49 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.74/2.49 ---------------------------------------- 5.74/2.49 5.74/2.49 (47) 5.74/2.49 YES 6.09/2.53 EOF