6.95/2.68 MAYBE 7.33/2.71 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 7.33/2.71 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.33/2.71 7.33/2.71 7.33/2.71 Left Termination of the query pattern 7.33/2.71 7.33/2.71 front(a,g) 7.33/2.71 7.33/2.71 w.r.t. the given Prolog program could not be shown: 7.33/2.71 7.33/2.71 (0) Prolog 7.33/2.71 (1) PrologToPiTRSProof [SOUND, 0 ms] 7.33/2.71 (2) PiTRS 7.33/2.71 (3) DependencyPairsProof [EQUIVALENT, 15 ms] 7.33/2.71 (4) PiDP 7.33/2.71 (5) DependencyGraphProof [EQUIVALENT, 7 ms] 7.33/2.71 (6) AND 7.33/2.71 (7) PiDP 7.33/2.71 (8) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (9) PiDP 7.33/2.71 (10) PiDPToQDPProof [SOUND, 8 ms] 7.33/2.71 (11) QDP 7.33/2.71 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.33/2.71 (13) YES 7.33/2.71 (14) PiDP 7.33/2.71 (15) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (16) PiDP 7.33/2.71 (17) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (18) QDP 7.33/2.71 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.33/2.71 (20) YES 7.33/2.71 (21) PiDP 7.33/2.71 (22) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (23) PiDP 7.33/2.71 (24) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (25) QDP 7.33/2.71 (26) PrologToPiTRSProof [SOUND, 0 ms] 7.33/2.71 (27) PiTRS 7.33/2.71 (28) DependencyPairsProof [EQUIVALENT, 3 ms] 7.33/2.71 (29) PiDP 7.33/2.71 (30) DependencyGraphProof [EQUIVALENT, 3 ms] 7.33/2.71 (31) AND 7.33/2.71 (32) PiDP 7.33/2.71 (33) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (34) PiDP 7.33/2.71 (35) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (36) QDP 7.33/2.71 (37) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.33/2.71 (38) YES 7.33/2.71 (39) PiDP 7.33/2.71 (40) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (41) PiDP 7.33/2.71 (42) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (43) QDP 7.33/2.71 (44) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.33/2.71 (45) YES 7.33/2.71 (46) PiDP 7.33/2.71 (47) UsableRulesProof [EQUIVALENT, 1 ms] 7.33/2.71 (48) PiDP 7.33/2.71 (49) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (50) QDP 7.33/2.71 (51) PrologToTRSTransformerProof [SOUND, 0 ms] 7.33/2.71 (52) QTRS 7.33/2.71 (53) QTRSRRRProof [EQUIVALENT, 147 ms] 7.33/2.71 (54) QTRS 7.33/2.71 (55) DependencyPairsProof [EQUIVALENT, 0 ms] 7.33/2.71 (56) QDP 7.33/2.71 (57) DependencyGraphProof [EQUIVALENT, 0 ms] 7.33/2.71 (58) AND 7.33/2.71 (59) QDP 7.33/2.71 (60) QDP 7.33/2.71 (61) PrologToDTProblemTransformerProof [SOUND, 103 ms] 7.33/2.71 (62) TRIPLES 7.33/2.71 (63) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] 7.33/2.71 (64) TRIPLES 7.33/2.71 (65) TriplesToPiDPProof [SOUND, 23 ms] 7.33/2.71 (66) PiDP 7.33/2.71 (67) DependencyGraphProof [EQUIVALENT, 0 ms] 7.33/2.71 (68) AND 7.33/2.71 (69) PiDP 7.33/2.71 (70) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (71) PiDP 7.33/2.71 (72) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (73) QDP 7.33/2.71 (74) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.33/2.71 (75) YES 7.33/2.71 (76) PiDP 7.33/2.71 (77) UsableRulesProof [EQUIVALENT, 0 ms] 7.33/2.71 (78) PiDP 7.33/2.71 (79) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (80) QDP 7.33/2.71 (81) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.33/2.71 (82) YES 7.33/2.71 (83) PiDP 7.33/2.71 (84) PiDPToQDPProof [SOUND, 0 ms] 7.33/2.71 (85) QDP 7.33/2.71 (86) PrologToIRSwTTransformerProof [SOUND, 53 ms] 7.33/2.71 (87) AND 7.33/2.71 (88) IRSwT 7.33/2.71 (89) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 7.33/2.71 (90) TRUE 7.33/2.71 (91) IRSwT 7.33/2.71 (92) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 7.33/2.71 (93) TRUE 7.33/2.71 (94) IRSwT 7.33/2.71 (95) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 7.33/2.71 (96) IRSwT 7.33/2.71 (97) IntTRSCompressionProof [EQUIVALENT, 19 ms] 7.33/2.71 (98) IRSwT 7.33/2.71 (99) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 7.33/2.71 (100) IRSwT 7.33/2.71 (101) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] 7.33/2.71 (102) IRSwT 7.33/2.71 (103) FilterProof [EQUIVALENT, 0 ms] 7.33/2.71 (104) IntTRS 7.33/2.71 (105) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] 7.33/2.71 (106) NO 7.33/2.71 7.33/2.71 7.33/2.71 ---------------------------------------- 7.33/2.71 7.33/2.71 (0) 7.33/2.71 Obligation: 7.33/2.71 Clauses: 7.33/2.71 7.33/2.71 front(void, []). 7.33/2.71 front(tree(X, void, void), .(X, [])). 7.33/2.71 front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs))). 7.33/2.71 app([], X, X). 7.33/2.71 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 7.33/2.71 7.33/2.71 7.33/2.71 Query: front(a,g) 7.33/2.71 ---------------------------------------- 7.33/2.71 7.33/2.71 (1) PrologToPiTRSProof (SOUND) 7.33/2.71 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 7.33/2.71 7.33/2.71 front_in_2: (f,b) (f,f) 7.33/2.71 7.33/2.71 app_in_3: (b,b,f) (b,b,b) 7.33/2.71 7.33/2.71 Transforming Prolog into the following Term Rewriting System: 7.33/2.71 7.33/2.71 Pi-finite rewrite system: 7.33/2.71 The TRS R consists of the following rules: 7.33/2.71 7.33/2.71 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.71 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.71 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.71 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.71 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.71 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.71 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.71 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.71 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.71 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.71 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.71 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.71 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.71 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.71 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.71 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.71 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.71 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.71 7.33/2.71 The argument filtering Pi contains the following mapping: 7.33/2.71 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.71 7.33/2.71 [] = [] 7.33/2.71 7.33/2.71 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.71 7.33/2.71 .(x1, x2) = .(x2) 7.33/2.71 7.33/2.71 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.71 7.33/2.71 front_in_aa(x1, x2) = front_in_aa 7.33/2.71 7.33/2.71 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.71 7.33/2.71 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.71 7.33/2.71 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.71 7.33/2.71 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.71 7.33/2.71 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.71 7.33/2.71 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.71 7.33/2.71 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.71 7.33/2.71 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.71 7.33/2.71 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.71 7.33/2.71 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.71 7.33/2.71 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.71 7.33/2.71 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.71 7.33/2.71 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.71 7.33/2.71 7.33/2.71 7.33/2.71 7.33/2.71 7.33/2.71 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 7.33/2.71 7.33/2.71 7.33/2.71 7.33/2.71 ---------------------------------------- 7.33/2.71 7.33/2.71 (2) 7.33/2.71 Obligation: 7.33/2.71 Pi-finite rewrite system: 7.33/2.71 The TRS R consists of the following rules: 7.33/2.71 7.33/2.71 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.71 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.71 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.71 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.72 7.33/2.72 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (3) DependencyPairsProof (EQUIVALENT) 7.33/2.72 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> U1_AG(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AA(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGA(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AG(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AG(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGG(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.72 7.33/2.72 FRONT_IN_AG(x1, x2) = FRONT_IN_AG(x2) 7.33/2.72 7.33/2.72 U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) 7.33/2.72 7.33/2.72 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.72 7.33/2.72 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.72 7.33/2.72 U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x2, x3, x5) 7.33/2.72 7.33/2.72 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.72 7.33/2.72 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) 7.33/2.72 7.33/2.72 U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x5) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_GGG(x1, x2, x3, x4, x5) = U4_GGG(x5) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (4) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> U1_AG(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AA(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGA(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AG(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AG(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGG(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.72 7.33/2.72 FRONT_IN_AG(x1, x2) = FRONT_IN_AG(x2) 7.33/2.72 7.33/2.72 U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) 7.33/2.72 7.33/2.72 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.72 7.33/2.72 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.72 7.33/2.72 U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x2, x3, x5) 7.33/2.72 7.33/2.72 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.72 7.33/2.72 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) 7.33/2.72 7.33/2.72 U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x5) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_GGG(x1, x2, x3, x4, x5) = U4_GGG(x5) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (5) DependencyGraphProof (EQUIVALENT) 7.33/2.72 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (6) 7.33/2.72 Complex Obligation (AND) 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (7) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (8) UsableRulesProof (EQUIVALENT) 7.33/2.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (9) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 R is empty. 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (10) PiDPToQDPProof (SOUND) 7.33/2.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (11) 7.33/2.72 Obligation: 7.33/2.72 Q DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 R is empty. 7.33/2.72 Q is empty. 7.33/2.72 We have to consider all (P,Q,R)-chains. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (12) QDPSizeChangeProof (EQUIVALENT) 7.33/2.72 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. 7.33/2.72 7.33/2.72 From the DPs we obtained the following set of size-change graphs: 7.33/2.72 *APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3 7.33/2.72 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (13) 7.33/2.72 YES 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (14) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.72 7.33/2.72 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (15) UsableRulesProof (EQUIVALENT) 7.33/2.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (16) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 R is empty. 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (17) PiDPToQDPProof (SOUND) 7.33/2.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (18) 7.33/2.72 Obligation: 7.33/2.72 Q DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.33/2.72 7.33/2.72 R is empty. 7.33/2.72 Q is empty. 7.33/2.72 We have to consider all (P,Q,R)-chains. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (19) QDPSizeChangeProof (EQUIVALENT) 7.33/2.72 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. 7.33/2.72 7.33/2.72 From the DPs we obtained the following set of size-change graphs: 7.33/2.72 *APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.33/2.72 The graph contains the following edges 1 > 1, 2 >= 2 7.33/2.72 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (20) 7.33/2.72 YES 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (21) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x5) 7.33/2.72 7.33/2.72 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.72 7.33/2.72 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (22) UsableRulesProof (EQUIVALENT) 7.33/2.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (23) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) 7.33/2.72 7.33/2.72 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.72 7.33/2.72 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (24) PiDPToQDPProof (SOUND) 7.33/2.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (25) 7.33/2.72 Obligation: 7.33/2.72 Q DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 U1_AA(front_out_aa(L, Ls)) -> FRONT_IN_AA 7.33/2.72 FRONT_IN_AA -> U1_AA(front_in_aa) 7.33/2.72 FRONT_IN_AA -> FRONT_IN_AA 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_aa -> front_out_aa(void, []) 7.33/2.72 front_in_aa -> front_out_aa(tree(void, void), .([])) 7.33/2.72 front_in_aa -> U1_aa(front_in_aa) 7.33/2.72 U1_aa(front_out_aa(L, Ls)) -> U2_aa(L, Ls, front_in_aa) 7.33/2.72 U2_aa(L, Ls, front_out_aa(R, Rs)) -> U3_aa(L, R, app_in_gga(Ls, Rs)) 7.33/2.72 U3_aa(L, R, app_out_gga(Xs)) -> front_out_aa(tree(L, R), Xs) 7.33/2.72 app_in_gga([], X) -> app_out_gga(X) 7.33/2.72 app_in_gga(.(Xs), Ys) -> U4_gga(app_in_gga(Xs, Ys)) 7.33/2.72 U4_gga(app_out_gga(Zs)) -> app_out_gga(.(Zs)) 7.33/2.72 7.33/2.72 The set Q consists of the following terms: 7.33/2.72 7.33/2.72 front_in_aa 7.33/2.72 U1_aa(x0) 7.33/2.72 U2_aa(x0, x1, x2) 7.33/2.72 U3_aa(x0, x1, x2) 7.33/2.72 app_in_gga(x0, x1) 7.33/2.72 U4_gga(x0) 7.33/2.72 7.33/2.72 We have to consider all (P,Q,R)-chains. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (26) PrologToPiTRSProof (SOUND) 7.33/2.72 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 7.33/2.72 7.33/2.72 front_in_2: (f,b) (f,f) 7.33/2.72 7.33/2.72 app_in_3: (b,b,f) (b,b,b) 7.33/2.72 7.33/2.72 Transforming Prolog into the following Term Rewriting System: 7.33/2.72 7.33/2.72 Pi-finite rewrite system: 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 7.33/2.72 7.33/2.72 7.33/2.72 7.33/2.72 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 7.33/2.72 7.33/2.72 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (27) 7.33/2.72 Obligation: 7.33/2.72 Pi-finite rewrite system: 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (28) DependencyPairsProof (EQUIVALENT) 7.33/2.72 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> U1_AG(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AA(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGA(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AG(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AG(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGG(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 FRONT_IN_AG(x1, x2) = FRONT_IN_AG(x2) 7.33/2.72 7.33/2.72 U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) 7.33/2.72 7.33/2.72 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.72 7.33/2.72 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.72 7.33/2.72 U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x2, x3, x5) 7.33/2.72 7.33/2.72 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.72 7.33/2.72 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_GGG(x1, x2, x3, x4, x5) = U4_GGG(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (29) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> U1_AG(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AG(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AA(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 U2_AA(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGA(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_AG(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U1_AG(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_AG(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 U2_AG(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> APP_IN_GGG(Ls, Rs, Xs) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U4_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 FRONT_IN_AG(x1, x2) = FRONT_IN_AG(x2) 7.33/2.72 7.33/2.72 U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) 7.33/2.72 7.33/2.72 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.72 7.33/2.72 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.72 7.33/2.72 U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x2, x3, x5) 7.33/2.72 7.33/2.72 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.72 7.33/2.72 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_GGG(x1, x2, x3, x4, x5) = U4_GGG(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (30) DependencyGraphProof (EQUIVALENT) 7.33/2.72 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (31) 7.33/2.72 Complex Obligation (AND) 7.33/2.72 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (32) 7.33/2.72 Obligation: 7.33/2.72 Pi DP problem: 7.33/2.72 The TRS P consists of the following rules: 7.33/2.72 7.33/2.72 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.72 7.33/2.72 The TRS R consists of the following rules: 7.33/2.72 7.33/2.72 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.72 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.72 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.72 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.72 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.72 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.72 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.72 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.72 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.72 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.72 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.72 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.72 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.72 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.72 7.33/2.72 The argument filtering Pi contains the following mapping: 7.33/2.72 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.72 7.33/2.72 [] = [] 7.33/2.72 7.33/2.72 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.72 7.33/2.72 .(x1, x2) = .(x2) 7.33/2.72 7.33/2.72 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.72 7.33/2.72 front_in_aa(x1, x2) = front_in_aa 7.33/2.72 7.33/2.72 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.72 7.33/2.72 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.72 7.33/2.72 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.72 7.33/2.72 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.72 7.33/2.72 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.72 7.33/2.72 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.72 7.33/2.72 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.72 7.33/2.72 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.72 7.33/2.72 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.72 7.33/2.72 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.72 7.33/2.72 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.72 7.33/2.72 7.33/2.72 We have to consider all (P,R,Pi)-chains 7.33/2.72 ---------------------------------------- 7.33/2.72 7.33/2.72 (33) UsableRulesProof (EQUIVALENT) 7.33/2.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.72 ---------------------------------------- 7.33/2.73 7.33/2.73 (34) 7.33/2.73 Obligation: 7.33/2.73 Pi DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.73 7.33/2.73 R is empty. 7.33/2.73 The argument filtering Pi contains the following mapping: 7.33/2.73 .(x1, x2) = .(x2) 7.33/2.73 7.33/2.73 APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3) 7.33/2.73 7.33/2.73 7.33/2.73 We have to consider all (P,R,Pi)-chains 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (35) PiDPToQDPProof (SOUND) 7.33/2.73 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (36) 7.33/2.73 Obligation: 7.33/2.73 Q DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.73 7.33/2.73 R is empty. 7.33/2.73 Q is empty. 7.33/2.73 We have to consider all (P,Q,R)-chains. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (37) QDPSizeChangeProof (EQUIVALENT) 7.33/2.73 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. 7.33/2.73 7.33/2.73 From the DPs we obtained the following set of size-change graphs: 7.33/2.73 *APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs) 7.33/2.73 The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3 7.33/2.73 7.33/2.73 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (38) 7.33/2.73 YES 7.33/2.73 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (39) 7.33/2.73 Obligation: 7.33/2.73 Pi DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.73 7.33/2.73 The TRS R consists of the following rules: 7.33/2.73 7.33/2.73 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.73 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.73 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.73 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.73 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.73 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.73 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.73 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.73 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.73 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.73 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.73 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.73 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.73 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.73 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.73 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.73 7.33/2.73 The argument filtering Pi contains the following mapping: 7.33/2.73 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.73 7.33/2.73 [] = [] 7.33/2.73 7.33/2.73 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.73 7.33/2.73 .(x1, x2) = .(x2) 7.33/2.73 7.33/2.73 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.73 7.33/2.73 front_in_aa(x1, x2) = front_in_aa 7.33/2.73 7.33/2.73 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.73 7.33/2.73 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.73 7.33/2.73 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.73 7.33/2.73 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.73 7.33/2.73 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.73 7.33/2.73 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.73 7.33/2.73 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.73 7.33/2.73 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.73 7.33/2.73 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.73 7.33/2.73 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.73 7.33/2.73 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.73 7.33/2.73 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.73 7.33/2.73 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.73 7.33/2.73 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.73 7.33/2.73 7.33/2.73 We have to consider all (P,R,Pi)-chains 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (40) UsableRulesProof (EQUIVALENT) 7.33/2.73 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (41) 7.33/2.73 Obligation: 7.33/2.73 Pi DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.33/2.73 7.33/2.73 R is empty. 7.33/2.73 The argument filtering Pi contains the following mapping: 7.33/2.73 .(x1, x2) = .(x2) 7.33/2.73 7.33/2.73 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.33/2.73 7.33/2.73 7.33/2.73 We have to consider all (P,R,Pi)-chains 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (42) PiDPToQDPProof (SOUND) 7.33/2.73 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (43) 7.33/2.73 Obligation: 7.33/2.73 Q DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.33/2.73 7.33/2.73 R is empty. 7.33/2.73 Q is empty. 7.33/2.73 We have to consider all (P,Q,R)-chains. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (44) QDPSizeChangeProof (EQUIVALENT) 7.33/2.73 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. 7.33/2.73 7.33/2.73 From the DPs we obtained the following set of size-change graphs: 7.33/2.73 *APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.33/2.73 The graph contains the following edges 1 > 1, 2 >= 2 7.33/2.73 7.33/2.73 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (45) 7.33/2.73 YES 7.33/2.73 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (46) 7.33/2.73 Obligation: 7.33/2.73 Pi DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.73 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.73 7.33/2.73 The TRS R consists of the following rules: 7.33/2.73 7.33/2.73 front_in_ag(void, []) -> front_out_ag(void, []) 7.33/2.73 front_in_ag(tree(X, void, void), .(X, [])) -> front_out_ag(tree(X, void, void), .(X, [])) 7.33/2.73 front_in_ag(tree(X1, L, R), Xs) -> U1_ag(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.73 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.73 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.73 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.73 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.73 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.73 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.73 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.73 U1_ag(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_ag(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.73 U2_ag(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_ag(X1, L, R, Xs, app_in_ggg(Ls, Rs, Xs)) 7.33/2.73 app_in_ggg([], X, X) -> app_out_ggg([], X, X) 7.33/2.73 app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U4_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs)) 7.33/2.73 U4_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs)) 7.33/2.73 U3_ag(X1, L, R, Xs, app_out_ggg(Ls, Rs, Xs)) -> front_out_ag(tree(X1, L, R), Xs) 7.33/2.73 7.33/2.73 The argument filtering Pi contains the following mapping: 7.33/2.73 front_in_ag(x1, x2) = front_in_ag(x2) 7.33/2.73 7.33/2.73 [] = [] 7.33/2.73 7.33/2.73 front_out_ag(x1, x2) = front_out_ag(x1, x2) 7.33/2.73 7.33/2.73 .(x1, x2) = .(x2) 7.33/2.73 7.33/2.73 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) 7.33/2.73 7.33/2.73 front_in_aa(x1, x2) = front_in_aa 7.33/2.73 7.33/2.73 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.73 7.33/2.73 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.73 7.33/2.73 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.73 7.33/2.73 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.73 7.33/2.73 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.73 7.33/2.73 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.73 7.33/2.73 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.73 7.33/2.73 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.73 7.33/2.73 U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x2, x4, x5, x6) 7.33/2.73 7.33/2.73 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5) 7.33/2.73 7.33/2.73 app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3) 7.33/2.73 7.33/2.73 app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3) 7.33/2.73 7.33/2.73 U4_ggg(x1, x2, x3, x4, x5) = U4_ggg(x2, x3, x4, x5) 7.33/2.73 7.33/2.73 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.73 7.33/2.73 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.73 7.33/2.73 7.33/2.73 We have to consider all (P,R,Pi)-chains 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (47) UsableRulesProof (EQUIVALENT) 7.33/2.73 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (48) 7.33/2.73 Obligation: 7.33/2.73 Pi DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 U1_AA(X1, L, R, Xs, front_out_aa(L, Ls)) -> FRONT_IN_AA(R, Rs) 7.33/2.73 FRONT_IN_AA(tree(X1, L, R), Xs) -> U1_AA(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 FRONT_IN_AA(tree(X1, L, R), Xs) -> FRONT_IN_AA(L, Ls) 7.33/2.73 7.33/2.73 The TRS R consists of the following rules: 7.33/2.73 7.33/2.73 front_in_aa(void, []) -> front_out_aa(void, []) 7.33/2.73 front_in_aa(tree(X, void, void), .(X, [])) -> front_out_aa(tree(X, void, void), .(X, [])) 7.33/2.73 front_in_aa(tree(X1, L, R), Xs) -> U1_aa(X1, L, R, Xs, front_in_aa(L, Ls)) 7.33/2.73 U1_aa(X1, L, R, Xs, front_out_aa(L, Ls)) -> U2_aa(X1, L, R, Xs, Ls, front_in_aa(R, Rs)) 7.33/2.73 U2_aa(X1, L, R, Xs, Ls, front_out_aa(R, Rs)) -> U3_aa(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 7.33/2.73 U3_aa(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(X1, L, R), Xs) 7.33/2.73 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.33/2.73 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.33/2.73 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.33/2.73 7.33/2.73 The argument filtering Pi contains the following mapping: 7.33/2.73 [] = [] 7.33/2.73 7.33/2.73 .(x1, x2) = .(x2) 7.33/2.73 7.33/2.73 front_in_aa(x1, x2) = front_in_aa 7.33/2.73 7.33/2.73 front_out_aa(x1, x2) = front_out_aa(x1, x2) 7.33/2.73 7.33/2.73 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.73 7.33/2.73 U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) 7.33/2.73 7.33/2.73 U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x2, x5, x6) 7.33/2.73 7.33/2.73 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x2, x3, x5) 7.33/2.73 7.33/2.73 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.33/2.73 7.33/2.73 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 7.33/2.73 7.33/2.73 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x3, x5) 7.33/2.73 7.33/2.73 FRONT_IN_AA(x1, x2) = FRONT_IN_AA 7.33/2.73 7.33/2.73 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.73 7.33/2.73 7.33/2.73 We have to consider all (P,R,Pi)-chains 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (49) PiDPToQDPProof (SOUND) 7.33/2.73 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (50) 7.33/2.73 Obligation: 7.33/2.73 Q DP problem: 7.33/2.73 The TRS P consists of the following rules: 7.33/2.73 7.33/2.73 U1_AA(front_out_aa(L, Ls)) -> FRONT_IN_AA 7.33/2.73 FRONT_IN_AA -> U1_AA(front_in_aa) 7.33/2.73 FRONT_IN_AA -> FRONT_IN_AA 7.33/2.73 7.33/2.73 The TRS R consists of the following rules: 7.33/2.73 7.33/2.73 front_in_aa -> front_out_aa(void, []) 7.33/2.73 front_in_aa -> front_out_aa(tree(void, void), .([])) 7.33/2.73 front_in_aa -> U1_aa(front_in_aa) 7.33/2.73 U1_aa(front_out_aa(L, Ls)) -> U2_aa(L, Ls, front_in_aa) 7.33/2.73 U2_aa(L, Ls, front_out_aa(R, Rs)) -> U3_aa(L, R, app_in_gga(Ls, Rs)) 7.33/2.73 U3_aa(L, R, app_out_gga(Ls, Rs, Xs)) -> front_out_aa(tree(L, R), Xs) 7.33/2.73 app_in_gga([], X) -> app_out_gga([], X, X) 7.33/2.73 app_in_gga(.(Xs), Ys) -> U4_gga(Xs, Ys, app_in_gga(Xs, Ys)) 7.33/2.73 U4_gga(Xs, Ys, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(Xs), Ys, .(Zs)) 7.33/2.73 7.33/2.73 The set Q consists of the following terms: 7.33/2.73 7.33/2.73 front_in_aa 7.33/2.73 U1_aa(x0) 7.33/2.73 U2_aa(x0, x1, x2) 7.33/2.73 U3_aa(x0, x1, x2) 7.33/2.73 app_in_gga(x0, x1) 7.33/2.73 U4_gga(x0, x1, x2) 7.33/2.73 7.33/2.73 We have to consider all (P,Q,R)-chains. 7.33/2.73 ---------------------------------------- 7.33/2.73 7.33/2.73 (51) PrologToTRSTransformerProof (SOUND) 7.33/2.73 Transformed Prolog program to TRS. 7.33/2.73 7.33/2.73 { 7.33/2.73 "root": 1, 7.33/2.73 "program": { 7.33/2.73 "directives": [], 7.33/2.73 "clauses": [ 7.33/2.73 [ 7.33/2.73 "(front (void) ([]))", 7.33/2.73 null 7.33/2.73 ], 7.33/2.73 [ 7.33/2.73 "(front (tree X (void) (void)) (. X ([])))", 7.33/2.73 null 7.33/2.73 ], 7.33/2.73 [ 7.33/2.73 "(front (tree X1 L R) Xs)", 7.33/2.73 "(',' (front L Ls) (',' (front R Rs) (app Ls Rs Xs)))" 7.33/2.73 ], 7.33/2.73 [ 7.33/2.73 "(app ([]) X X)", 7.33/2.73 null 7.33/2.73 ], 7.33/2.73 [ 7.33/2.73 "(app (. X Xs) Ys (. X Zs))", 7.33/2.73 "(app Xs Ys Zs)" 7.33/2.73 ] 7.33/2.73 ] 7.33/2.73 }, 7.33/2.73 "graph": { 7.33/2.73 "nodes": { 7.33/2.73 "type": "Nodes", 7.33/2.73 "271": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X21"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "273": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(',' (front T23 X22) (app T22 X22 T19))" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": ["T19"], 7.33/2.73 "free": ["X22"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "350": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "351": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(app T60 T61 X68)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X68"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "352": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "276": { 7.33/2.73 "goal": [ 7.33/2.73 { 7.33/2.73 "clause": 0, 7.33/2.73 "scope": 2, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 }, 7.33/2.73 { 7.33/2.73 "clause": 1, 7.33/2.73 "scope": 2, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 }, 7.33/2.73 { 7.33/2.73 "clause": 2, 7.33/2.73 "scope": 2, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 } 7.33/2.73 ], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X21"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "353": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(front T23 X22)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X22"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "354": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(app T65 T64 T19)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": ["T19"], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "278": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": 0, 7.33/2.73 "scope": 2, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X21"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "355": { 7.33/2.73 "goal": [ 7.33/2.73 { 7.33/2.73 "clause": 3, 7.33/2.73 "scope": 4, 7.33/2.73 "term": "(app T65 T64 T19)" 7.33/2.73 }, 7.33/2.73 { 7.33/2.73 "clause": 4, 7.33/2.73 "scope": 4, 7.33/2.73 "term": "(app T65 T64 T19)" 7.33/2.73 } 7.33/2.73 ], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": ["T19"], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "279": { 7.33/2.73 "goal": [ 7.33/2.73 { 7.33/2.73 "clause": 1, 7.33/2.73 "scope": 2, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 }, 7.33/2.73 { 7.33/2.73 "clause": 2, 7.33/2.73 "scope": 2, 7.33/2.73 "term": "(front T20 X21)" 7.33/2.73 } 7.33/2.73 ], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X21"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "356": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": 3, 7.33/2.73 "scope": 4, 7.33/2.73 "term": "(app T65 T64 T19)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": ["T19"], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "357": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": 4, 7.33/2.73 "scope": 4, 7.33/2.73 "term": "(app T65 T64 T19)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": ["T19"], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "358": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(true)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "359": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "99": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 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7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "104": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "346": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": 3, 7.33/2.73 "scope": 3, 7.33/2.73 "term": "(app T43 T42 X46)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X46"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "347": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": 4, 7.33/2.73 "scope": 3, 7.33/2.73 "term": "(app T43 T42 X46)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": ["X46"], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "348": { 7.33/2.73 "goal": [{ 7.33/2.73 "clause": -1, 7.33/2.73 "scope": -1, 7.33/2.73 "term": "(true)" 7.33/2.73 }], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "349": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.73 "arithmetic": { 7.33/2.73 "type": "PlainIntegerRelationState", 7.33/2.73 "relations": [] 7.33/2.73 }, 7.33/2.73 "ground": [], 7.33/2.73 "free": [], 7.33/2.73 "exprvars": [] 7.33/2.73 } 7.33/2.73 }, 7.33/2.73 "84": { 7.33/2.73 "goal": [], 7.33/2.73 "kb": { 7.33/2.73 "nonunifying": [], 7.33/2.73 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "edges": [ 7.33/2.74 { 7.33/2.74 "from": 1, 7.33/2.74 "to": 5, 7.33/2.74 "label": "CASE" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 5, 7.33/2.74 "to": 18, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 5, 7.33/2.74 "to": 19, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 18, 7.33/2.74 "to": 75, 7.33/2.74 "label": "EVAL with clause\nfront(void, []).\nand substitutionT1 -> void,\nT2 -> []" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 18, 7.33/2.74 "to": 84, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 19, 7.33/2.74 "to": 100, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 19, 7.33/2.74 "to": 101, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 75, 7.33/2.74 "to": 99, 7.33/2.74 "label": "SUCCESS" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 100, 7.33/2.74 "to": 102, 7.33/2.74 "label": "EVAL with clause\nfront(tree(X6, void, void), .(X6, [])).\nand substitutionX6 -> T7,\nT1 -> tree(T7, void, void),\nT2 -> .(T7, [])" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 100, 7.33/2.74 "to": 103, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 101, 7.33/2.74 "to": 265, 7.33/2.74 "label": "EVAL with clause\nfront(tree(X17, X18, X19), X20) :- ','(front(X18, X21), ','(front(X19, X22), app(X21, X22, X20))).\nand substitutionX17 -> T16,\nX18 -> T20,\nX19 -> T21,\nT1 -> tree(T16, T20, T21),\nT2 -> T19,\nX20 -> T19,\nT17 -> T20,\nT18 -> T21" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 101, 7.33/2.74 "to": 266, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 102, 7.33/2.74 "to": 104, 7.33/2.74 "label": "SUCCESS" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 265, 7.33/2.74 "to": 271, 7.33/2.74 "label": "SPLIT 1" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 265, 7.33/2.74 "to": 273, 7.33/2.74 "label": "SPLIT 2\nreplacements:X21 -> T22,\nT21 -> T23" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 271, 7.33/2.74 "to": 276, 7.33/2.74 "label": "CASE" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 273, 7.33/2.74 "to": 353, 7.33/2.74 "label": "SPLIT 1" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 273, 7.33/2.74 "to": 354, 7.33/2.74 "label": "SPLIT 2\nreplacements:X22 -> T64,\nT22 -> T65" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 276, 7.33/2.74 "to": 278, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 276, 7.33/2.74 "to": 279, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 278, 7.33/2.74 "to": 280, 7.33/2.74 "label": "EVAL with clause\nfront(void, []).\nand substitutionT20 -> void,\nX21 -> []" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 278, 7.33/2.74 "to": 281, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 279, 7.33/2.74 "to": 286, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 279, 7.33/2.74 "to": 287, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 280, 7.33/2.74 "to": 283, 7.33/2.74 "label": "SUCCESS" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 286, 7.33/2.74 "to": 288, 7.33/2.74 "label": "EVAL with clause\nfront(tree(X27, void, void), .(X27, [])).\nand substitutionX27 -> T28,\nT20 -> tree(T28, void, void),\nX21 -> .(T28, [])" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 286, 7.33/2.74 "to": 289, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 287, 7.33/2.74 "to": 293, 7.33/2.74 "label": "EVAL with clause\nfront(tree(X40, X41, X42), X43) :- ','(front(X41, X44), ','(front(X42, X45), app(X44, X45, X43))).\nand substitutionX40 -> T35,\nX41 -> T38,\nX42 -> T39,\nT20 -> tree(T35, T38, T39),\nX21 -> X46,\nX43 -> X46,\nT36 -> T38,\nT37 -> T39" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 287, 7.33/2.74 "to": 294, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 288, 7.33/2.74 "to": 290, 7.33/2.74 "label": "SUCCESS" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 293, 7.33/2.74 "to": 297, 7.33/2.74 "label": "SPLIT 1" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 293, 7.33/2.74 "to": 298, 7.33/2.74 "label": "SPLIT 2\nreplacements:X44 -> T40,\nT39 -> T41" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 297, 7.33/2.74 "to": 271, 7.33/2.74 "label": "INSTANCE with matching:\nT20 -> T38\nX21 -> X44" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 298, 7.33/2.74 "to": 331, 7.33/2.74 "label": "SPLIT 1" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 298, 7.33/2.74 "to": 332, 7.33/2.74 "label": "SPLIT 2\nreplacements:X45 -> T42,\nT40 -> T43" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 331, 7.33/2.74 "to": 271, 7.33/2.74 "label": "INSTANCE with matching:\nT20 -> T41\nX21 -> X45" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 332, 7.33/2.74 "to": 333, 7.33/2.74 "label": "CASE" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 333, 7.33/2.74 "to": 346, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 333, 7.33/2.74 "to": 347, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 346, 7.33/2.74 "to": 348, 7.33/2.74 "label": "EVAL with clause\napp([], X53, X53).\nand substitutionT43 -> [],\nT42 -> T50,\nX53 -> T50,\nX46 -> T50" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 346, 7.33/2.74 "to": 349, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 347, 7.33/2.74 "to": 351, 7.33/2.74 "label": "EVAL with clause\napp(.(X64, X65), X66, .(X64, X67)) :- app(X65, X66, X67).\nand substitutionX64 -> T57,\nX65 -> T60,\nT43 -> .(T57, T60),\nT42 -> T61,\nX66 -> T61,\nX67 -> X68,\nX46 -> .(T57, X68),\nT58 -> T60,\nT59 -> T61" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 347, 7.33/2.74 "to": 352, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 348, 7.33/2.74 "to": 350, 7.33/2.74 "label": "SUCCESS" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 351, 7.33/2.74 "to": 332, 7.33/2.74 "label": "INSTANCE with matching:\nT43 -> T60\nT42 -> T61\nX46 -> X68" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 353, 7.33/2.74 "to": 271, 7.33/2.74 "label": "INSTANCE with matching:\nT20 -> T23\nX21 -> X22" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 354, 7.33/2.74 "to": 355, 7.33/2.74 "label": "CASE" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 355, 7.33/2.74 "to": 356, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 355, 7.33/2.74 "to": 357, 7.33/2.74 "label": "PARALLEL" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 356, 7.33/2.74 "to": 358, 7.33/2.74 "label": "EVAL with clause\napp([], X77, X77).\nand substitutionT65 -> [],\nT64 -> T72,\nX77 -> T72,\nT19 -> T72" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 356, 7.33/2.74 "to": 359, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 357, 7.33/2.74 "to": 361, 7.33/2.74 "label": "EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T81,\nX87 -> T85,\nT65 -> .(T81, T85),\nT64 -> T86,\nX88 -> T86,\nX89 -> T84,\nT19 -> .(T81, T84),\nT82 -> T85,\nT83 -> T86" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 357, 7.33/2.74 "to": 362, 7.33/2.74 "label": "EVAL-BACKTRACK" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 358, 7.33/2.74 "to": 360, 7.33/2.74 "label": "SUCCESS" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "from": 361, 7.33/2.74 "to": 354, 7.33/2.74 "label": "INSTANCE with matching:\nT65 -> T85\nT64 -> T86\nT19 -> T84" 7.33/2.74 } 7.33/2.74 ], 7.33/2.74 "type": "Graph" 7.33/2.74 } 7.33/2.74 } 7.33/2.74 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (52) 7.33/2.74 Obligation: 7.33/2.74 Q restricted rewrite system: 7.33/2.74 The TRS R consists of the following rules: 7.33/2.74 7.33/2.74 f1_in([]) -> f1_out1 7.33/2.74 f1_in(.(T7, [])) -> f1_out1 7.33/2.74 f1_in(T19) -> U1(f265_in(T19), T19) 7.33/2.74 U1(f265_out1(X21, X22), T19) -> f1_out1 7.33/2.74 f271_in -> f271_out1 7.33/2.74 f271_in -> U2(f293_in) 7.33/2.74 U2(f293_out1) -> f271_out1 7.33/2.74 f332_in -> f332_out1 7.33/2.74 f332_in -> U3(f332_in) 7.33/2.74 U3(f332_out1) -> f332_out1 7.33/2.74 f354_in(T72) -> f354_out1([], T72) 7.33/2.74 f354_in(.(T81, T84)) -> U4(f354_in(T84), .(T81, T84)) 7.33/2.74 U4(f354_out1(T85, T86), .(T81, T84)) -> f354_out1(.(T81, T85), T86) 7.33/2.74 f265_in(T19) -> U5(f271_in, T19) 7.33/2.74 U5(f271_out1, T19) -> U6(f273_in(T19), T19) 7.33/2.74 U6(f273_out1(T22, X22), T19) -> f265_out1(T22, X22) 7.33/2.74 f273_in(T19) -> U7(f271_in, T19) 7.33/2.74 U7(f271_out1, T19) -> U8(f354_in(T19), T19) 7.33/2.74 U8(f354_out1(T65, T64), T19) -> f273_out1(T65, T64) 7.33/2.74 f293_in -> U9(f271_in) 7.33/2.74 U9(f271_out1) -> U10(f298_in) 7.33/2.74 U10(f298_out1) -> f293_out1 7.33/2.74 f298_in -> U11(f271_in) 7.33/2.74 U11(f271_out1) -> U12(f332_in) 7.33/2.74 U12(f332_out1) -> f298_out1 7.33/2.74 7.33/2.74 Q is empty. 7.33/2.74 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (53) QTRSRRRProof (EQUIVALENT) 7.33/2.74 Used ordering: 7.33/2.74 f1_in/1(YES) 7.33/2.74 []/0) 7.33/2.74 f1_out1/0) 7.33/2.74 ./2(YES,YES) 7.33/2.74 U1/2(YES,YES) 7.33/2.74 f265_in/1(YES) 7.33/2.74 f265_out1/2(YES,YES) 7.33/2.74 f271_in/0) 7.33/2.74 f271_out1/0) 7.33/2.74 U2/1)YES( 7.33/2.74 f293_in/0) 7.33/2.74 f293_out1/0) 7.33/2.74 f332_in/0) 7.33/2.74 f332_out1/0) 7.33/2.74 U3/1)YES( 7.33/2.74 f354_in/1(YES) 7.33/2.74 f354_out1/2(YES,YES) 7.33/2.74 U4/2(YES,YES) 7.33/2.74 U5/2(YES,YES) 7.33/2.74 U6/2(YES,YES) 7.33/2.74 f273_in/1(YES) 7.33/2.74 f273_out1/2(YES,YES) 7.33/2.74 U7/2(YES,YES) 7.33/2.74 U8/2(YES,YES) 7.33/2.74 U9/1)YES( 7.33/2.74 U10/1)YES( 7.33/2.74 f298_in/0) 7.33/2.74 f298_out1/0) 7.33/2.74 U11/1)YES( 7.33/2.74 U12/1)YES( 7.33/2.74 7.33/2.74 Quasi precedence: 7.33/2.74 f1_in_1 > U1_2 7.33/2.74 f1_in_1 > f265_in_1 > [U5_2, f273_in_1] > U6_2 > f265_out1_2 > f1_out1 7.33/2.74 f1_in_1 > f265_in_1 > [U5_2, f273_in_1] > U7_2 > [f271_in, f271_out1, f293_in, f293_out1, f332_in, f332_out1, f273_out1_2, U8_2, f298_in, f298_out1] > f354_in_1 > [] 7.33/2.74 f1_in_1 > f265_in_1 > [U5_2, f273_in_1] > U7_2 > [f271_in, f271_out1, f293_in, f293_out1, f332_in, f332_out1, f273_out1_2, U8_2, f298_in, f298_out1] > f354_in_1 > [f354_out1_2, U4_2] > ._2 7.33/2.74 7.33/2.74 7.33/2.74 Status: 7.33/2.74 f1_in_1: multiset status 7.33/2.74 []: multiset status 7.33/2.74 f1_out1: multiset status 7.33/2.74 ._2: multiset status 7.33/2.74 U1_2: multiset status 7.33/2.74 f265_in_1: multiset status 7.33/2.74 f265_out1_2: [1,2] 7.33/2.74 f271_in: multiset status 7.33/2.74 f271_out1: multiset status 7.33/2.74 f293_in: multiset status 7.33/2.74 f293_out1: multiset status 7.33/2.74 f332_in: multiset status 7.33/2.74 f332_out1: multiset status 7.33/2.74 f354_in_1: multiset status 7.33/2.74 f354_out1_2: [2,1] 7.33/2.74 U4_2: [1,2] 7.33/2.74 U5_2: multiset status 7.33/2.74 U6_2: multiset status 7.33/2.74 f273_in_1: multiset status 7.33/2.74 f273_out1_2: multiset status 7.33/2.74 U7_2: multiset status 7.33/2.74 U8_2: multiset status 7.33/2.74 f298_in: multiset status 7.33/2.74 f298_out1: multiset status 7.33/2.74 7.33/2.74 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 7.33/2.74 7.33/2.74 f1_in([]) -> f1_out1 7.33/2.74 f1_in(.(T7, [])) -> f1_out1 7.33/2.74 f1_in(T19) -> U1(f265_in(T19), T19) 7.33/2.74 U1(f265_out1(X21, X22), T19) -> f1_out1 7.33/2.74 f354_in(T72) -> f354_out1([], T72) 7.33/2.74 f354_in(.(T81, T84)) -> U4(f354_in(T84), .(T81, T84)) 7.33/2.74 U4(f354_out1(T85, T86), .(T81, T84)) -> f354_out1(.(T81, T85), T86) 7.33/2.74 f265_in(T19) -> U5(f271_in, T19) 7.33/2.74 U5(f271_out1, T19) -> U6(f273_in(T19), T19) 7.33/2.74 U6(f273_out1(T22, X22), T19) -> f265_out1(T22, X22) 7.33/2.74 f273_in(T19) -> U7(f271_in, T19) 7.33/2.74 U7(f271_out1, T19) -> U8(f354_in(T19), T19) 7.33/2.74 U8(f354_out1(T65, T64), T19) -> f273_out1(T65, T64) 7.33/2.74 7.33/2.74 7.33/2.74 7.33/2.74 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (54) 7.33/2.74 Obligation: 7.33/2.74 Q restricted rewrite system: 7.33/2.74 The TRS R consists of the following rules: 7.33/2.74 7.33/2.74 f271_in -> f271_out1 7.33/2.74 f271_in -> U2(f293_in) 7.33/2.74 U2(f293_out1) -> f271_out1 7.33/2.74 f332_in -> f332_out1 7.33/2.74 f332_in -> U3(f332_in) 7.33/2.74 U3(f332_out1) -> f332_out1 7.33/2.74 f293_in -> U9(f271_in) 7.33/2.74 U9(f271_out1) -> U10(f298_in) 7.33/2.74 U10(f298_out1) -> f293_out1 7.33/2.74 f298_in -> U11(f271_in) 7.33/2.74 U11(f271_out1) -> U12(f332_in) 7.33/2.74 U12(f332_out1) -> f298_out1 7.33/2.74 7.33/2.74 Q is empty. 7.33/2.74 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (55) DependencyPairsProof (EQUIVALENT) 7.33/2.74 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (56) 7.33/2.74 Obligation: 7.33/2.74 Q DP problem: 7.33/2.74 The TRS P consists of the following rules: 7.33/2.74 7.33/2.74 F271_IN -> U2^1(f293_in) 7.33/2.74 F271_IN -> F293_IN 7.33/2.74 F332_IN -> U3^1(f332_in) 7.33/2.74 F332_IN -> F332_IN 7.33/2.74 F293_IN -> U9^1(f271_in) 7.33/2.74 F293_IN -> F271_IN 7.33/2.74 U9^1(f271_out1) -> U10^1(f298_in) 7.33/2.74 U9^1(f271_out1) -> F298_IN 7.33/2.74 F298_IN -> U11^1(f271_in) 7.33/2.74 F298_IN -> F271_IN 7.33/2.74 U11^1(f271_out1) -> U12^1(f332_in) 7.33/2.74 U11^1(f271_out1) -> F332_IN 7.33/2.74 7.33/2.74 The TRS R consists of the following rules: 7.33/2.74 7.33/2.74 f271_in -> f271_out1 7.33/2.74 f271_in -> U2(f293_in) 7.33/2.74 U2(f293_out1) -> f271_out1 7.33/2.74 f332_in -> f332_out1 7.33/2.74 f332_in -> U3(f332_in) 7.33/2.74 U3(f332_out1) -> f332_out1 7.33/2.74 f293_in -> U9(f271_in) 7.33/2.74 U9(f271_out1) -> U10(f298_in) 7.33/2.74 U10(f298_out1) -> f293_out1 7.33/2.74 f298_in -> U11(f271_in) 7.33/2.74 U11(f271_out1) -> U12(f332_in) 7.33/2.74 U12(f332_out1) -> f298_out1 7.33/2.74 7.33/2.74 Q is empty. 7.33/2.74 We have to consider all minimal (P,Q,R)-chains. 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (57) DependencyGraphProof (EQUIVALENT) 7.33/2.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes. 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (58) 7.33/2.74 Complex Obligation (AND) 7.33/2.74 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (59) 7.33/2.74 Obligation: 7.33/2.74 Q DP problem: 7.33/2.74 The TRS P consists of the following rules: 7.33/2.74 7.33/2.74 F332_IN -> F332_IN 7.33/2.74 7.33/2.74 The TRS R consists of the following rules: 7.33/2.74 7.33/2.74 f271_in -> f271_out1 7.33/2.74 f271_in -> U2(f293_in) 7.33/2.74 U2(f293_out1) -> f271_out1 7.33/2.74 f332_in -> f332_out1 7.33/2.74 f332_in -> U3(f332_in) 7.33/2.74 U3(f332_out1) -> f332_out1 7.33/2.74 f293_in -> U9(f271_in) 7.33/2.74 U9(f271_out1) -> U10(f298_in) 7.33/2.74 U10(f298_out1) -> f293_out1 7.33/2.74 f298_in -> U11(f271_in) 7.33/2.74 U11(f271_out1) -> U12(f332_in) 7.33/2.74 U12(f332_out1) -> f298_out1 7.33/2.74 7.33/2.74 Q is empty. 7.33/2.74 We have to consider all minimal (P,Q,R)-chains. 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (60) 7.33/2.74 Obligation: 7.33/2.74 Q DP problem: 7.33/2.74 The TRS P consists of the following rules: 7.33/2.74 7.33/2.74 F271_IN -> F293_IN 7.33/2.74 F293_IN -> U9^1(f271_in) 7.33/2.74 U9^1(f271_out1) -> F298_IN 7.33/2.74 F298_IN -> F271_IN 7.33/2.74 F293_IN -> F271_IN 7.33/2.74 7.33/2.74 The TRS R consists of the following rules: 7.33/2.74 7.33/2.74 f271_in -> f271_out1 7.33/2.74 f271_in -> U2(f293_in) 7.33/2.74 U2(f293_out1) -> f271_out1 7.33/2.74 f332_in -> f332_out1 7.33/2.74 f332_in -> U3(f332_in) 7.33/2.74 U3(f332_out1) -> f332_out1 7.33/2.74 f293_in -> U9(f271_in) 7.33/2.74 U9(f271_out1) -> U10(f298_in) 7.33/2.74 U10(f298_out1) -> f293_out1 7.33/2.74 f298_in -> U11(f271_in) 7.33/2.74 U11(f271_out1) -> U12(f332_in) 7.33/2.74 U12(f332_out1) -> f298_out1 7.33/2.74 7.33/2.74 Q is empty. 7.33/2.74 We have to consider all minimal (P,Q,R)-chains. 7.33/2.74 ---------------------------------------- 7.33/2.74 7.33/2.74 (61) PrologToDTProblemTransformerProof (SOUND) 7.33/2.74 Built DT problem from termination graph DT10. 7.33/2.74 7.33/2.74 { 7.33/2.74 "root": 16, 7.33/2.74 "program": { 7.33/2.74 "directives": [], 7.33/2.74 "clauses": [ 7.33/2.74 [ 7.33/2.74 "(front (void) ([]))", 7.33/2.74 null 7.33/2.74 ], 7.33/2.74 [ 7.33/2.74 "(front (tree X (void) (void)) (. X ([])))", 7.33/2.74 null 7.33/2.74 ], 7.33/2.74 [ 7.33/2.74 "(front (tree X1 L R) Xs)", 7.33/2.74 "(',' (front L Ls) (',' (front R Rs) (app Ls Rs Xs)))" 7.33/2.74 ], 7.33/2.74 [ 7.33/2.74 "(app ([]) X X)", 7.33/2.74 null 7.33/2.74 ], 7.33/2.74 [ 7.33/2.74 "(app (. X Xs) Ys (. X Zs))", 7.33/2.74 "(app Xs Ys Zs)" 7.33/2.74 ] 7.33/2.74 ] 7.33/2.74 }, 7.33/2.74 "graph": { 7.33/2.74 "nodes": { 7.33/2.74 "type": "Nodes", 7.33/2.74 "470": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": -1, 7.33/2.74 "scope": -1, 7.33/2.74 "term": "(app (. T144 ([])) T143 (. T117 ([])))" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": ["T117"], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "471": { 7.33/2.74 "goal": [ 7.33/2.74 { 7.33/2.74 "clause": 3, 7.33/2.74 "scope": 10, 7.33/2.74 "term": "(app (. T144 ([])) T143 (. T117 ([])))" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "clause": 4, 7.33/2.74 "scope": 10, 7.33/2.74 "term": "(app (. T144 ([])) T143 (. T117 ([])))" 7.33/2.74 } 7.33/2.74 ], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": ["T117"], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "472": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": 4, 7.33/2.74 "scope": 10, 7.33/2.74 "term": "(app (. T144 ([])) T143 (. T117 ([])))" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": ["T117"], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "110": { 7.33/2.74 "goal": [ 7.33/2.74 { 7.33/2.74 "clause": 1, 7.33/2.74 "scope": 1, 7.33/2.74 "term": "(front T1 T2)" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "clause": 2, 7.33/2.74 "scope": 1, 7.33/2.74 "term": "(front T1 T2)" 7.33/2.74 } 7.33/2.74 ], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [[ 7.33/2.74 "(front T1 T2)", 7.33/2.74 "(front (void) ([]))" 7.33/2.74 ]], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": ["T2"], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "473": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": -1, 7.33/2.74 "scope": -1, 7.33/2.74 "term": "(app ([]) T151 ([]))" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "474": { 7.33/2.74 "goal": [], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "112": { 7.33/2.74 "goal": [ 7.33/2.74 { 7.33/2.74 "clause": 1, 7.33/2.74 "scope": 1, 7.33/2.74 "term": "(front T1 ([]))" 7.33/2.74 }, 7.33/2.74 { 7.33/2.74 "clause": 2, 7.33/2.74 "scope": 1, 7.33/2.74 "term": "(front T1 ([]))" 7.33/2.74 } 7.33/2.74 ], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "475": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": -1, 7.33/2.74 "scope": -1, 7.33/2.74 "term": "(',' (',' (front T163 X179) (',' (front T164 X180) (app X179 X180 X181))) (',' (front T165 X129) (app X181 X129 (. T117 ([])))))" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": ["T117"], 7.33/2.74 "free": [ 7.33/2.74 "X129", 7.33/2.74 "X181", 7.33/2.74 "X179", 7.33/2.74 "X180" 7.33/2.74 ], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "113": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": 2, 7.33/2.74 "scope": 1, 7.33/2.74 "term": "(front T1 ([]))" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "476": { 7.33/2.74 "goal": [], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": [], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "477": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": -1, 7.33/2.74 "scope": -1, 7.33/2.74 "term": "(front T163 X179)" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [], 7.33/2.74 "intvars": {}, 7.33/2.74 "arithmetic": { 7.33/2.74 "type": "PlainIntegerRelationState", 7.33/2.74 "relations": [] 7.33/2.74 }, 7.33/2.74 "ground": [], 7.33/2.74 "free": ["X179"], 7.33/2.74 "exprvars": [] 7.33/2.74 } 7.33/2.74 }, 7.33/2.74 "510": { 7.33/2.74 "goal": [{ 7.33/2.74 "clause": 2, 7.33/2.74 "scope": 12, 7.33/2.74 "term": "(',' (front T215 X215) (',' (front T216 X216) (app X215 X216 T214)))" 7.33/2.74 }], 7.33/2.74 "kb": { 7.33/2.74 "nonunifying": [ 7.33/2.74 [ 7.33/2.74 "(front T1 T214)", 7.33/2.74 "(front (void) ([]))" 7.33/2.74 ], 7.33/2.74 [ 7.33/2.74 "(front T1 T214)", 7.33/2.74 "(front (tree X119 (void) (void)) (. 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T117 ([])))" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T117"], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "461": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(true)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "462": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "463": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "464": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "465": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": 1, 7.33/2.76 "scope": 8, 7.33/2.76 "term": "(',' (front T118 X128) (',' (front T119 X129) (app X128 X129 (. T117 ([])))))" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T117"], 7.33/2.76 "free": [ 7.33/2.76 "X128", 7.33/2.76 "X129" 7.33/2.76 ], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "466": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": 2, 7.33/2.76 "scope": 8, 7.33/2.76 "term": "(',' (front T118 X128) (',' (front T119 X129) (app X128 X129 (. T117 ([])))))" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T117"], 7.33/2.76 "free": [ 7.33/2.76 "X128", 7.33/2.76 "X129" 7.33/2.76 ], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "467": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(',' (front T141 X129) (app (. T142 ([])) X129 (. T117 ([]))))" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T117"], 7.33/2.76 "free": ["X129"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "500": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(front T217 X216)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": ["X216"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "468": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "501": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(app ([]) T218 T214)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [ 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (void) ([]))" 7.33/2.76 ], 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (tree X119 (void) (void)) (. X119 ([])))" 7.33/2.76 ] 7.33/2.76 ], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T214"], 7.33/2.76 "free": ["X119"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "469": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(front T141 X129)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": ["X129"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "502": { 7.33/2.76 "goal": [ 7.33/2.76 { 7.33/2.76 "clause": 3, 7.33/2.76 "scope": 13, 7.33/2.76 "term": "(app ([]) T218 T214)" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "clause": 4, 7.33/2.76 "scope": 13, 7.33/2.76 "term": "(app ([]) T218 T214)" 7.33/2.76 } 7.33/2.76 ], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [ 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (void) ([]))" 7.33/2.76 ], 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (tree X119 (void) (void)) (. X119 ([])))" 7.33/2.76 ] 7.33/2.76 ], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T214"], 7.33/2.76 "free": ["X119"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "503": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": 3, 7.33/2.76 "scope": 13, 7.33/2.76 "term": "(app ([]) T218 T214)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [ 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (void) ([]))" 7.33/2.76 ], 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (tree X119 (void) (void)) (. X119 ([])))" 7.33/2.76 ] 7.33/2.76 ], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T214"], 7.33/2.76 "free": ["X119"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "108": { 7.33/2.76 "goal": [ 7.33/2.76 { 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(true)" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "clause": 1, 7.33/2.76 "scope": 1, 7.33/2.76 "term": "(front T1 ([]))" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "clause": 2, 7.33/2.76 "scope": 1, 7.33/2.76 "term": "(front T1 ([]))" 7.33/2.76 } 7.33/2.76 ], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "504": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": 4, 7.33/2.76 "scope": 13, 7.33/2.76 "term": "(app ([]) T218 T214)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [ 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (void) ([]))" 7.33/2.76 ], 7.33/2.76 [ 7.33/2.76 "(front T219 T214)", 7.33/2.76 "(front (tree X119 (void) (void)) (. X119 ([])))" 7.33/2.76 ] 7.33/2.76 ], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T214"], 7.33/2.76 "free": ["X119"], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "505": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": -1, 7.33/2.76 "scope": -1, 7.33/2.76 "term": "(true)" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "506": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "507": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "508": { 7.33/2.76 "goal": [], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": [], 7.33/2.76 "free": [], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "509": { 7.33/2.76 "goal": [{ 7.33/2.76 "clause": 1, 7.33/2.76 "scope": 12, 7.33/2.76 "term": "(',' (front T215 X215) (',' (front T216 X216) (app X215 X216 T214)))" 7.33/2.76 }], 7.33/2.76 "kb": { 7.33/2.76 "nonunifying": [ 7.33/2.76 [ 7.33/2.76 "(front T1 T214)", 7.33/2.76 "(front (void) ([]))" 7.33/2.76 ], 7.33/2.76 [ 7.33/2.76 "(front T1 T214)", 7.33/2.76 "(front (tree X119 (void) (void)) (. X119 ([])))" 7.33/2.76 ] 7.33/2.76 ], 7.33/2.76 "intvars": {}, 7.33/2.76 "arithmetic": { 7.33/2.76 "type": "PlainIntegerRelationState", 7.33/2.76 "relations": [] 7.33/2.76 }, 7.33/2.76 "ground": ["T214"], 7.33/2.76 "free": [ 7.33/2.76 "X119", 7.33/2.76 "X215", 7.33/2.76 "X216" 7.33/2.76 ], 7.33/2.76 "exprvars": [] 7.33/2.76 } 7.33/2.76 } 7.33/2.76 }, 7.33/2.76 "edges": [ 7.33/2.76 { 7.33/2.76 "from": 16, 7.33/2.76 "to": 17, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 17, 7.33/2.76 "to": 108, 7.33/2.76 "label": "EVAL with clause\nfront(void, []).\nand substitutionT1 -> void,\nT2 -> []" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 17, 7.33/2.76 "to": 110, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 108, 7.33/2.76 "to": 112, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 110, 7.33/2.76 "to": 430, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X119, void, void), .(X119, [])).\nand substitutionX119 -> T109,\nT1 -> tree(T109, void, void),\nT2 -> .(T109, [])" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 110, 7.33/2.76 "to": 431, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 112, 7.33/2.76 "to": 113, 7.33/2.76 "label": "BACKTRACK\nfor clause: front(tree(X, void, void), .(X, []))because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 113, 7.33/2.76 "to": 116, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X7, X8, X9), X10) :- ','(front(X8, X11), ','(front(X9, X12), app(X11, X12, X10))).\nand substitutionX7 -> T7,\nX8 -> T10,\nX9 -> T11,\nT1 -> tree(T7, T10, T11),\nX10 -> [],\nT8 -> T10,\nT9 -> T11" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 113, 7.33/2.76 "to": 118, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 116, 7.33/2.76 "to": 127, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 127, 7.33/2.76 "to": 128, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 127, 7.33/2.76 "to": 129, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 128, 7.33/2.76 "to": 142, 7.33/2.76 "label": "EVAL with clause\nfront(void, []).\nand substitutionT10 -> void,\nX11 -> [],\nT11 -> T12" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 128, 7.33/2.76 "to": 145, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 129, 7.33/2.76 "to": 394, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 129, 7.33/2.76 "to": 395, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 142, 7.33/2.76 "to": 299, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 142, 7.33/2.76 "to": 300, 7.33/2.76 "label": "SPLIT 2\nreplacements:X12 -> T13" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 299, 7.33/2.76 "to": 301, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 300, 7.33/2.76 "to": 387, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 301, 7.33/2.76 "to": 302, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 301, 7.33/2.76 "to": 303, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 302, 7.33/2.76 "to": 304, 7.33/2.76 "label": "EVAL with clause\nfront(void, []).\nand substitutionT12 -> void,\nX12 -> []" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 302, 7.33/2.76 "to": 305, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 303, 7.33/2.76 "to": 307, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 303, 7.33/2.76 "to": 308, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 304, 7.33/2.76 "to": 306, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 307, 7.33/2.76 "to": 309, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X17, void, void), .(X17, [])).\nand substitutionX17 -> T18,\nT12 -> tree(T18, void, void),\nX12 -> .(T18, [])" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 307, 7.33/2.76 "to": 310, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 308, 7.33/2.76 "to": 317, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X30, X31, X32), X33) :- ','(front(X31, X34), ','(front(X32, X35), app(X34, X35, X33))).\nand substitutionX30 -> T25,\nX31 -> T28,\nX32 -> T29,\nT12 -> tree(T25, T28, T29),\nX12 -> X36,\nX33 -> X36,\nT26 -> T28,\nT27 -> T29" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 308, 7.33/2.76 "to": 319, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 309, 7.33/2.76 "to": 311, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 317, 7.33/2.76 "to": 327, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 317, 7.33/2.76 "to": 328, 7.33/2.76 "label": "SPLIT 2\nreplacements:X34 -> T30,\nT29 -> T31" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 327, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T28\nX12 -> X34" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 328, 7.33/2.76 "to": 329, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 328, 7.33/2.76 "to": 330, 7.33/2.76 "label": "SPLIT 2\nreplacements:X35 -> T32,\nT30 -> T33" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 329, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T31\nX12 -> X35" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 330, 7.33/2.76 "to": 379, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 379, 7.33/2.76 "to": 380, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 379, 7.33/2.76 "to": 381, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 380, 7.33/2.76 "to": 382, 7.33/2.76 "label": "EVAL with clause\napp([], X43, X43).\nand substitutionT33 -> [],\nT32 -> T40,\nX43 -> T40,\nX36 -> T40" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 380, 7.33/2.76 "to": 383, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 381, 7.33/2.76 "to": 385, 7.33/2.76 "label": "EVAL with clause\napp(.(X54, X55), X56, .(X54, X57)) :- app(X55, X56, X57).\nand substitutionX54 -> T47,\nX55 -> T50,\nT33 -> .(T47, T50),\nT32 -> T51,\nX56 -> T51,\nX57 -> X58,\nX36 -> .(T47, X58),\nT48 -> T50,\nT49 -> T51" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 381, 7.33/2.76 "to": 386, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 382, 7.33/2.76 "to": 384, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 385, 7.33/2.76 "to": 330, 7.33/2.76 "label": "INSTANCE with matching:\nT33 -> T50\nT32 -> T51\nX36 -> X58" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 387, 7.33/2.76 "to": 388, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 387, 7.33/2.76 "to": 389, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 388, 7.33/2.76 "to": 390, 7.33/2.76 "label": "EVAL with clause\napp([], X67, X67).\nand substitutionT13 -> [],\nX67 -> [],\nT60 -> []" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 388, 7.33/2.76 "to": 391, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 389, 7.33/2.76 "to": 393, 7.33/2.76 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 390, 7.33/2.76 "to": 392, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 394, 7.33/2.76 "to": 396, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X76, void, void), .(X76, [])).\nand substitutionX76 -> T67,\nT10 -> tree(T67, void, void),\nX11 -> .(T67, []),\nT11 -> T66,\nT65 -> T67" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 394, 7.33/2.76 "to": 397, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 395, 7.33/2.76 "to": 413, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X96, X97, X98), X99) :- ','(front(X97, X100), ','(front(X98, X101), app(X100, X101, X99))).\nand substitutionX96 -> T78,\nX97 -> T81,\nX98 -> T82,\nT10 -> tree(T78, T81, T82),\nX11 -> X102,\nX99 -> X102,\nT79 -> T81,\nT80 -> T82,\nT11 -> T83" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 395, 7.33/2.76 "to": 414, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 396, 7.33/2.76 "to": 400, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 396, 7.33/2.76 "to": 401, 7.33/2.76 "label": "SPLIT 2\nreplacements:X12 -> T68,\nT67 -> T69" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 400, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T66" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 401, 7.33/2.76 "to": 405, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 405, 7.33/2.76 "to": 406, 7.33/2.76 "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 406, 7.33/2.76 "to": 409, 7.33/2.76 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 413, 7.33/2.76 "to": 415, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 413, 7.33/2.76 "to": 416, 7.33/2.76 "label": "SPLIT 2\nreplacements:X100 -> T84,\nT82 -> T85,\nT83 -> T86" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 415, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T81\nX12 -> X100" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 416, 7.33/2.76 "to": 417, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 416, 7.33/2.76 "to": 418, 7.33/2.76 "label": "SPLIT 2\nreplacements:X101 -> T87,\nT84 -> T88,\nT86 -> T89" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 417, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T85\nX12 -> X101" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 418, 7.33/2.76 "to": 419, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 418, 7.33/2.76 "to": 420, 7.33/2.76 "label": "SPLIT 2\nreplacements:X102 -> T92,\nT89 -> T93" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 419, 7.33/2.76 "to": 330, 7.33/2.76 "label": "INSTANCE with matching:\nT33 -> T88\nT32 -> T87\nX36 -> X102" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 420, 7.33/2.76 "to": 421, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 420, 7.33/2.76 "to": 422, 7.33/2.76 "label": "SPLIT 2\nreplacements:X12 -> T96,\nT92 -> T97" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 421, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T93" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 422, 7.33/2.76 "to": 423, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 423, 7.33/2.76 "to": 424, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 423, 7.33/2.76 "to": 425, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 424, 7.33/2.76 "to": 426, 7.33/2.76 "label": "EVAL with clause\napp([], X113, X113).\nand substitutionT97 -> [],\nT96 -> [],\nX113 -> [],\nT104 -> []" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 424, 7.33/2.76 "to": 427, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 425, 7.33/2.76 "to": 429, 7.33/2.76 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 426, 7.33/2.76 "to": 428, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 430, 7.33/2.76 "to": 432, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 431, 7.33/2.76 "to": 493, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X211, X212, X213), X214) :- ','(front(X212, X215), ','(front(X213, X216), app(X215, X216, X214))).\nand substitutionX211 -> T211,\nX212 -> T215,\nX213 -> T216,\nT1 -> tree(T211, T215, T216),\nT2 -> T214,\nX214 -> T214,\nT212 -> T215,\nT213 -> T216" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 431, 7.33/2.76 "to": 494, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 432, 7.33/2.76 "to": 433, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X124, X125, X126), X127) :- ','(front(X125, X128), ','(front(X126, X129), app(X128, X129, X127))).\nand substitutionX124 -> T114,\nX125 -> T118,\nX126 -> T119,\nT1 -> tree(T114, T118, T119),\nT109 -> T117,\nX127 -> .(T117, []),\nT115 -> T118,\nT116 -> T119" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 432, 7.33/2.76 "to": 434, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 433, 7.33/2.76 "to": 435, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 435, 7.33/2.76 "to": 436, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 435, 7.33/2.76 "to": 437, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 436, 7.33/2.76 "to": 438, 7.33/2.76 "label": "EVAL with clause\nfront(void, []).\nand substitutionT118 -> void,\nX128 -> [],\nT119 -> T120" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 436, 7.33/2.76 "to": 439, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 437, 7.33/2.76 "to": 465, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 437, 7.33/2.76 "to": 466, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 438, 7.33/2.76 "to": 456, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 438, 7.33/2.76 "to": 457, 7.33/2.76 "label": "SPLIT 2\nreplacements:X129 -> T121" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 456, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T120\nX12 -> X129" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 457, 7.33/2.76 "to": 458, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 458, 7.33/2.76 "to": 459, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 458, 7.33/2.76 "to": 460, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 459, 7.33/2.76 "to": 461, 7.33/2.76 "label": "EVAL with clause\napp([], X136, X136).\nand substitutionT121 -> .(T135, []),\nX136 -> .(T135, []),\nT117 -> T135,\nT134 -> .(T135, [])" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 459, 7.33/2.76 "to": 462, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 460, 7.33/2.76 "to": 464, 7.33/2.76 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 461, 7.33/2.76 "to": 463, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 465, 7.33/2.76 "to": 467, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X145, void, void), .(X145, [])).\nand substitutionX145 -> T142,\nT118 -> tree(T142, void, void),\nX128 -> .(T142, []),\nT119 -> T141,\nT140 -> T142" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 465, 7.33/2.76 "to": 468, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 466, 7.33/2.76 "to": 475, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X175, X176, X177), X178) :- ','(front(X176, X179), ','(front(X177, X180), app(X179, X180, X178))).\nand substitutionX175 -> T160,\nX176 -> T163,\nX177 -> T164,\nT118 -> tree(T160, T163, T164),\nX128 -> X181,\nX178 -> X181,\nT161 -> T163,\nT162 -> T164,\nT119 -> T165" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 466, 7.33/2.76 "to": 476, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 467, 7.33/2.76 "to": 469, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 467, 7.33/2.76 "to": 470, 7.33/2.76 "label": "SPLIT 2\nreplacements:X129 -> T143,\nT142 -> T144" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 469, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T141\nX12 -> X129" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 470, 7.33/2.76 "to": 471, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 471, 7.33/2.76 "to": 472, 7.33/2.76 "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 472, 7.33/2.76 "to": 473, 7.33/2.76 "label": "EVAL with clause\napp(.(X157, X158), X159, .(X157, X160)) :- app(X158, X159, X160).\nand substitutionT144 -> T149,\nX157 -> T149,\nX158 -> [],\nT143 -> T151,\nX159 -> T151,\nT117 -> T149,\nX160 -> [],\nT150 -> T151" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 472, 7.33/2.76 "to": 474, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 473, 7.33/2.76 "to": 300, 7.33/2.76 "label": "INSTANCE with matching:\nT13 -> T151" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 475, 7.33/2.76 "to": 477, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 475, 7.33/2.76 "to": 478, 7.33/2.76 "label": "SPLIT 2\nreplacements:X179 -> T166,\nT164 -> T167,\nT165 -> T168" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 477, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T163\nX12 -> X179" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 478, 7.33/2.76 "to": 479, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 478, 7.33/2.76 "to": 480, 7.33/2.76 "label": "SPLIT 2\nreplacements:X180 -> T169,\nT166 -> T170,\nT168 -> T171" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 479, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T167\nX12 -> X180" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 480, 7.33/2.76 "to": 481, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 480, 7.33/2.76 "to": 482, 7.33/2.76 "label": "SPLIT 2\nreplacements:X181 -> T174,\nT171 -> T175" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 481, 7.33/2.76 "to": 330, 7.33/2.76 "label": "INSTANCE with matching:\nT33 -> T170\nT32 -> T169\nX36 -> X181" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 482, 7.33/2.76 "to": 483, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 482, 7.33/2.76 "to": 484, 7.33/2.76 "label": "SPLIT 2\nreplacements:X129 -> T178,\nT174 -> T179" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 483, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T175\nX12 -> X129" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 484, 7.33/2.76 "to": 485, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 485, 7.33/2.76 "to": 486, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 485, 7.33/2.76 "to": 487, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 486, 7.33/2.76 "to": 488, 7.33/2.76 "label": "EVAL with clause\napp([], X192, X192).\nand substitutionT179 -> [],\nT178 -> .(T193, []),\nX192 -> .(T193, []),\nT117 -> T193,\nT192 -> .(T193, [])" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 486, 7.33/2.76 "to": 489, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 487, 7.33/2.76 "to": 491, 7.33/2.76 "label": "EVAL with clause\napp(.(X201, X202), X203, .(X201, X204)) :- app(X202, X203, X204).\nand substitutionX201 -> T200,\nX202 -> T203,\nT179 -> .(T200, T203),\nT178 -> T204,\nX203 -> T204,\nT117 -> T200,\nX204 -> [],\nT201 -> T203,\nT202 -> T204" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 487, 7.33/2.76 "to": 492, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 488, 7.33/2.76 "to": 490, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 491, 7.33/2.76 "to": 422, 7.33/2.76 "label": "INSTANCE with matching:\nT97 -> T203\nT96 -> T204" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 493, 7.33/2.76 "to": 495, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 495, 7.33/2.76 "to": 496, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 495, 7.33/2.76 "to": 497, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 496, 7.33/2.76 "to": 498, 7.33/2.76 "label": "EVAL with clause\nfront(void, []).\nand substitutionT215 -> void,\nX215 -> [],\nT216 -> T217" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 496, 7.33/2.76 "to": 499, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 497, 7.33/2.76 "to": 509, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 497, 7.33/2.76 "to": 510, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 498, 7.33/2.76 "to": 500, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 498, 7.33/2.76 "to": 501, 7.33/2.76 "label": "SPLIT 2\nreplacements:X216 -> T218,\nT1 -> T219" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 500, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T217\nX12 -> X216" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 501, 7.33/2.76 "to": 502, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 502, 7.33/2.76 "to": 503, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 502, 7.33/2.76 "to": 504, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 503, 7.33/2.76 "to": 505, 7.33/2.76 "label": "EVAL with clause\napp([], X223, X223).\nand substitutionT218 -> T226,\nX223 -> T226,\nT214 -> T226" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 503, 7.33/2.76 "to": 506, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 504, 7.33/2.76 "to": 508, 7.33/2.76 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 505, 7.33/2.76 "to": 507, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 509, 7.33/2.76 "to": 511, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X232, void, void), .(X232, [])).\nand substitutionX232 -> T233,\nT215 -> tree(T233, void, void),\nX215 -> .(T233, []),\nT216 -> T232,\nT231 -> T233" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 509, 7.33/2.76 "to": 512, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 510, 7.33/2.76 "to": 526, 7.33/2.76 "label": "EVAL with clause\nfront(tree(X271, X272, X273), X274) :- ','(front(X272, X275), ','(front(X273, X276), app(X275, X276, X274))).\nand substitutionX271 -> T260,\nX272 -> T263,\nX273 -> T264,\nT215 -> tree(T260, T263, T264),\nX215 -> X277,\nX274 -> X277,\nT261 -> T263,\nT262 -> T264,\nT216 -> T265" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 510, 7.33/2.76 "to": 527, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 511, 7.33/2.76 "to": 513, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 511, 7.33/2.76 "to": 514, 7.33/2.76 "label": "SPLIT 2\nreplacements:X216 -> T234,\nT233 -> T235,\nT1 -> T236" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 513, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T232\nX12 -> X216" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 514, 7.33/2.76 "to": 515, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 515, 7.33/2.76 "to": 516, 7.33/2.76 "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 516, 7.33/2.76 "to": 517, 7.33/2.76 "label": "EVAL with clause\napp(.(X244, X245), X246, .(X244, X247)) :- app(X245, X246, X247).\nand substitutionT235 -> T243,\nX244 -> T243,\nX245 -> [],\nT234 -> T246,\nX246 -> T246,\nX247 -> T245,\nT214 -> .(T243, T245),\nT244 -> T246" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 516, 7.33/2.76 "to": 518, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 517, 7.33/2.76 "to": 519, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 519, 7.33/2.76 "to": 520, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 519, 7.33/2.76 "to": 521, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 520, 7.33/2.76 "to": 522, 7.33/2.76 "label": "EVAL with clause\napp([], X254, X254).\nand substitutionT246 -> T253,\nX254 -> T253,\nT245 -> T253" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 520, 7.33/2.76 "to": 523, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 521, 7.33/2.76 "to": 525, 7.33/2.76 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 522, 7.33/2.76 "to": 524, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 526, 7.33/2.76 "to": 528, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 526, 7.33/2.76 "to": 529, 7.33/2.76 "label": "SPLIT 2\nreplacements:X275 -> T266,\nT264 -> T267,\nT265 -> T268,\nT1 -> T269" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 528, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T263\nX12 -> X275" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 529, 7.33/2.76 "to": 530, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 529, 7.33/2.76 "to": 531, 7.33/2.76 "label": "SPLIT 2\nreplacements:X276 -> T270,\nT266 -> T271,\nT268 -> T272,\nT269 -> T273" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 530, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T267\nX12 -> X276" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 531, 7.33/2.76 "to": 532, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 531, 7.33/2.76 "to": 533, 7.33/2.76 "label": "SPLIT 2\nreplacements:X277 -> T276,\nT272 -> T277,\nT273 -> T278" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 532, 7.33/2.76 "to": 330, 7.33/2.76 "label": "INSTANCE with matching:\nT33 -> T271\nT32 -> T270\nX36 -> X277" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 533, 7.33/2.76 "to": 534, 7.33/2.76 "label": "SPLIT 1" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 533, 7.33/2.76 "to": 535, 7.33/2.76 "label": "SPLIT 2\nreplacements:X216 -> T281,\nT276 -> T282,\nT278 -> T283" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 534, 7.33/2.76 "to": 299, 7.33/2.76 "label": "INSTANCE with matching:\nT12 -> T277\nX12 -> X216" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 535, 7.33/2.76 "to": 536, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 536, 7.33/2.76 "to": 537, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 536, 7.33/2.76 "to": 538, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 537, 7.33/2.76 "to": 539, 7.33/2.76 "label": "EVAL with clause\napp([], X288, X288).\nand substitutionT282 -> [],\nT281 -> T290,\nX288 -> T290,\nT214 -> T290" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 537, 7.33/2.76 "to": 540, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 538, 7.33/2.76 "to": 542, 7.33/2.76 "label": "EVAL with clause\napp(.(X297, X298), X299, .(X297, X300)) :- app(X298, X299, X300).\nand substitutionX297 -> T299,\nX298 -> T303,\nT282 -> .(T299, T303),\nT281 -> T304,\nX299 -> T304,\nX300 -> T302,\nT214 -> .(T299, T302),\nT300 -> T303,\nT301 -> T304" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 538, 7.33/2.76 "to": 543, 7.33/2.76 "label": "EVAL-BACKTRACK" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 539, 7.33/2.76 "to": 541, 7.33/2.76 "label": "SUCCESS" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 542, 7.33/2.76 "to": 544, 7.33/2.76 "label": "CASE" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 544, 7.33/2.76 "to": 545, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 544, 7.33/2.76 "to": 546, 7.33/2.76 "label": "PARALLEL" 7.33/2.76 }, 7.33/2.76 { 7.33/2.76 "from": 545, 7.33/2.76 "to": 547, 7.33/2.76 "label": "EVAL with clause\napp([], X307, X307).\nand substitutionT303 -> [],\nT304 -> T311,\nX307 -> T311,\nT302 -> T311" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 545, 7.33/2.77 "to": 548, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 546, 7.33/2.77 "to": 550, 7.33/2.77 "label": "EVAL with clause\napp(.(X316, X317), X318, .(X316, X319)) :- app(X317, X318, X319).\nand substitutionX316 -> T320,\nX317 -> T324,\nT303 -> .(T320, T324),\nT304 -> T325,\nX318 -> T325,\nX319 -> T323,\nT302 -> .(T320, T323),\nT321 -> T324,\nT322 -> T325" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 546, 7.33/2.77 "to": 551, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 547, 7.33/2.77 "to": 549, 7.33/2.77 "label": "SUCCESS" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 550, 7.33/2.77 "to": 552, 7.33/2.77 "label": "CASE" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 552, 7.33/2.77 "to": 553, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 552, 7.33/2.77 "to": 554, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 553, 7.33/2.77 "to": 555, 7.33/2.77 "label": "EVAL with clause\napp([], X326, X326).\nand substitutionT324 -> [],\nT325 -> T332,\nX326 -> T332,\nT323 -> T332" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 553, 7.33/2.77 "to": 556, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 554, 7.33/2.77 "to": 558, 7.33/2.77 "label": "EVAL with clause\napp(.(X335, X336), X337, .(X335, X338)) :- app(X336, X337, X338).\nand substitutionX335 -> T341,\nX336 -> T345,\nT324 -> .(T341, T345),\nT325 -> T346,\nX337 -> T346,\nX338 -> T344,\nT323 -> .(T341, T344),\nT342 -> T345,\nT343 -> T346" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 554, 7.33/2.77 "to": 559, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 555, 7.33/2.77 "to": 557, 7.33/2.77 "label": "SUCCESS" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 558, 7.33/2.77 "to": 550, 7.33/2.77 "label": "INSTANCE with matching:\nT324 -> T345\nT325 -> T346\nT323 -> T344" 7.33/2.77 } 7.33/2.77 ], 7.33/2.77 "type": "Graph" 7.33/2.77 } 7.33/2.77 } 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (62) 7.33/2.77 Obligation: 7.33/2.77 Triples: 7.33/2.77 7.33/2.77 frontA(tree(X1, X2, X3), X4) :- frontA(X2, X5). 7.33/2.77 frontA(tree(X1, X2, X3), X4) :- ','(frontcA(X2, X5), frontA(X3, X6)). 7.33/2.77 frontA(tree(X1, X2, X3), X4) :- ','(frontcA(X2, X5), ','(frontcA(X3, X6), appB(X5, X6, X4))). 7.33/2.77 appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4). 7.33/2.77 appE(.(X1, X2), X3, .(X1, X4)) :- appE(X2, X3, X4). 7.33/2.77 frontF(tree(X1, void, X2), []) :- frontA(X2, X3). 7.33/2.77 frontF(tree(X1, void, X2), []) :- ','(frontcA(X2, X3), appC(X3)). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), []) :- frontA(X3, X4). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- frontA(X3, X6). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), frontA(X4, X7)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), ','(frontcA(X4, X7), appB(X6, X7, X8))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), ','(frontcA(X4, X7), ','(appcB(X6, X7, X8), frontA(X5, X9)))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), ','(frontcA(X4, X7), ','(appcB(X6, X7, X8), ','(frontcA(X5, X9), appD(X8, X9))))). 7.33/2.77 frontF(tree(X1, void, X2), .(X3, [])) :- frontA(X2, X4). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), .(X4, [])) :- frontA(X3, X5). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), .(X2, [])) :- ','(frontcA(X3, X4), appC(X4)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- frontA(X3, X7). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), frontA(X4, X8)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), appB(X7, X8, X9))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), ','(appcB(X7, X8, X9), frontA(X5, X10)))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), ','(appcB(X7, X8, .(X6, X9)), ','(frontcA(X5, X10), appD(X9, X10))))). 7.33/2.77 frontF(tree(X1, void, X2), X3) :- frontA(X2, X4). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), X4) :- frontA(X3, X5). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- frontA(X3, X7). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(frontcA(X3, X7), frontA(X4, X8)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), appB(X7, X8, X9))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), ','(appcB(X7, X8, X9), frontA(X5, X10)))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, .(X7, X8))) :- ','(frontcA(X3, X9), ','(frontcA(X4, X10), ','(appcB(X9, X10, .(X6, .(X7, X11))), ','(frontcA(X5, X12), appE(X11, X12, X8))))). 7.33/2.77 7.33/2.77 Clauses: 7.33/2.77 7.33/2.77 frontcA(void, []). 7.33/2.77 frontcA(tree(X1, void, void), .(X1, [])). 7.33/2.77 frontcA(tree(X1, X2, X3), X4) :- ','(frontcA(X2, X5), ','(frontcA(X3, X6), appcB(X5, X6, X4))). 7.33/2.77 appcB([], X1, X1). 7.33/2.77 appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4). 7.33/2.77 appcC([]). 7.33/2.77 appcD([], []). 7.33/2.77 appcE([], X1, X1). 7.33/2.77 appcE(.(X1, X2), X3, .(X1, X4)) :- appcE(X2, X3, X4). 7.33/2.77 7.33/2.77 Afs: 7.33/2.77 7.33/2.77 frontF(x1, x2) = frontF(x2) 7.33/2.77 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (63) UndefinedPredicateInTriplesTransformerProof (SOUND) 7.33/2.77 Deleted triples and predicates having undefined goals [DT09]. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (64) 7.33/2.77 Obligation: 7.33/2.77 Triples: 7.33/2.77 7.33/2.77 frontA(tree(X1, X2, X3), X4) :- frontA(X2, X5). 7.33/2.77 frontA(tree(X1, X2, X3), X4) :- ','(frontcA(X2, X5), frontA(X3, X6)). 7.33/2.77 frontA(tree(X1, X2, X3), X4) :- ','(frontcA(X2, X5), ','(frontcA(X3, X6), appB(X5, X6, X4))). 7.33/2.77 appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4). 7.33/2.77 appE(.(X1, X2), X3, .(X1, X4)) :- appE(X2, X3, X4). 7.33/2.77 frontF(tree(X1, void, X2), []) :- frontA(X2, X3). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), []) :- frontA(X3, X4). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- frontA(X3, X6). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), frontA(X4, X7)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), ','(frontcA(X4, X7), appB(X6, X7, X8))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), []) :- ','(frontcA(X3, X6), ','(frontcA(X4, X7), ','(appcB(X6, X7, X8), frontA(X5, X9)))). 7.33/2.77 frontF(tree(X1, void, X2), .(X3, [])) :- frontA(X2, X4). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), .(X4, [])) :- frontA(X3, X5). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- frontA(X3, X7). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), frontA(X4, X8)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), appB(X7, X8, X9))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), ','(appcB(X7, X8, X9), frontA(X5, X10)))). 7.33/2.77 frontF(tree(X1, void, X2), X3) :- frontA(X2, X4). 7.33/2.77 frontF(tree(X1, tree(X2, void, void), X3), X4) :- frontA(X3, X5). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- frontA(X3, X7). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(frontcA(X3, X7), frontA(X4, X8)). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), appB(X7, X8, X9))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(frontcA(X3, X7), ','(frontcA(X4, X8), ','(appcB(X7, X8, X9), frontA(X5, X10)))). 7.33/2.77 frontF(tree(X1, tree(X2, X3, X4), X5), .(X6, .(X7, X8))) :- ','(frontcA(X3, X9), ','(frontcA(X4, X10), ','(appcB(X9, X10, .(X6, .(X7, X11))), ','(frontcA(X5, X12), appE(X11, X12, X8))))). 7.33/2.77 7.33/2.77 Clauses: 7.33/2.77 7.33/2.77 frontcA(void, []). 7.33/2.77 frontcA(tree(X1, void, void), .(X1, [])). 7.33/2.77 frontcA(tree(X1, X2, X3), X4) :- ','(frontcA(X2, X5), ','(frontcA(X3, X6), appcB(X5, X6, X4))). 7.33/2.77 appcB([], X1, X1). 7.33/2.77 appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4). 7.33/2.77 appcC([]). 7.33/2.77 appcD([], []). 7.33/2.77 appcE([], X1, X1). 7.33/2.77 appcE(.(X1, X2), X3, .(X1, X4)) :- appcE(X2, X3, X4). 7.33/2.77 7.33/2.77 Afs: 7.33/2.77 7.33/2.77 frontF(x1, x2) = frontF(x2) 7.33/2.77 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (65) TriplesToPiDPProof (SOUND) 7.33/2.77 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 7.33/2.77 7.33/2.77 frontF_in_2: (f,b) 7.33/2.77 7.33/2.77 frontA_in_2: (f,f) 7.33/2.77 7.33/2.77 frontcA_in_2: (f,f) 7.33/2.77 7.33/2.77 appcB_in_3: (b,b,f) 7.33/2.77 7.33/2.77 appB_in_3: (b,b,f) 7.33/2.77 7.33/2.77 appE_in_3: (b,b,b) 7.33/2.77 7.33/2.77 Transforming TRIPLES into the following Term Rewriting System: 7.33/2.77 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), []) -> U8_AG(X1, X2, frontA_in_aa(X2, X3)) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), []) -> FRONTA_IN_AA(X2, X3) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> U1_AA(X1, X2, X3, X4, frontA_in_aa(X2, X5)) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> FRONTA_IN_AA(X2, X5) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> U2_AA(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U3_AA(X1, X2, X3, X4, frontA_in_aa(X3, X6)) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> FRONTA_IN_AA(X3, X6) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U4_AA(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U4_AA(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U5_AA(X1, X2, X3, X4, appB_in_gga(X5, X6, X4)) 7.33/2.77 U4_AA(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> APPB_IN_GGA(X5, X6, X4) 7.33/2.77 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> U6_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4)) 7.33/2.77 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), []) -> U9_AG(X1, X2, X3, frontA_in_aa(X3, X4)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), []) -> FRONTA_IN_AA(X3, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), []) -> U10_AG(X1, X2, X3, X4, X5, frontA_in_aa(X3, X6)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), []) -> FRONTA_IN_AA(X3, X6) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), []) -> U11_AG(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U11_AG(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U12_AG(X1, X2, X3, X4, X5, frontA_in_aa(X4, X7)) 7.33/2.77 U11_AG(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> FRONTA_IN_AA(X4, X7) 7.33/2.77 U11_AG(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U13_AG(X1, X2, X3, X4, X5, X6, frontcA_in_aa(X4, X7)) 7.33/2.77 U13_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X4, X7)) -> U14_AG(X1, X2, X3, X4, X5, appB_in_gga(X6, X7, X8)) 7.33/2.77 U13_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X4, X7)) -> APPB_IN_GGA(X6, X7, X8) 7.33/2.77 U13_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X4, X7)) -> U15_AG(X1, X2, X3, X4, X5, appcB_in_gga(X6, X7, X8)) 7.33/2.77 U15_AG(X1, X2, X3, X4, X5, appcB_out_gga(X6, X7, X8)) -> U16_AG(X1, X2, X3, X4, X5, frontA_in_aa(X5, X9)) 7.33/2.77 U15_AG(X1, X2, X3, X4, X5, appcB_out_gga(X6, X7, X8)) -> FRONTA_IN_AA(X5, X9) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), .(X3, [])) -> U17_AG(X1, X2, X3, frontA_in_aa(X2, X4)) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), .(X3, [])) -> FRONTA_IN_AA(X2, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), .(X4, [])) -> U18_AG(X1, X2, X3, X4, frontA_in_aa(X3, X5)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), .(X4, [])) -> FRONTA_IN_AA(X3, X5) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) -> U19_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X3, X7)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) -> FRONTA_IN_AA(X3, X7) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) -> U20_AG(X1, X2, X3, X4, X5, X6, frontcA_in_aa(X3, X7)) 7.33/2.77 U20_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U21_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X4, X8)) 7.33/2.77 U20_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> FRONTA_IN_AA(X4, X8) 7.33/2.77 U20_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_in_aa(X4, X8)) 7.33/2.77 U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U23_AG(X1, X2, X3, X4, X5, X6, appB_in_gga(X7, X8, X9)) 7.33/2.77 U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> APPB_IN_GGA(X7, X8, X9) 7.33/2.77 U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U24_AG(X1, X2, X3, X4, X5, X6, appcB_in_gga(X7, X8, X9)) 7.33/2.77 U24_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> U25_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X5, X10)) 7.33/2.77 U24_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> FRONTA_IN_AA(X5, X10) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), X3) -> U26_AG(X1, X2, X3, frontA_in_aa(X2, X4)) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), X3) -> FRONTA_IN_AA(X2, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), X4) -> U27_AG(X1, X2, X3, X4, frontA_in_aa(X3, X5)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), X4) -> FRONTA_IN_AA(X3, X5) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U28_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X3, X7)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> FRONTA_IN_AA(X3, X7) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U29_AG(X1, X2, X3, X4, X5, X6, frontcA_in_aa(X3, X7)) 7.33/2.77 U29_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U30_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X4, X8)) 7.33/2.77 U29_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> FRONTA_IN_AA(X4, X8) 7.33/2.77 U29_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_in_aa(X4, X8)) 7.33/2.77 U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U32_AG(X1, X2, X3, X4, X5, X6, appB_in_gga(X7, X8, X9)) 7.33/2.77 U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> APPB_IN_GGA(X7, X8, X9) 7.33/2.77 U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U33_AG(X1, X2, X3, X4, X5, X6, appcB_in_gga(X7, X8, X9)) 7.33/2.77 U33_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> U34_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X5, X10)) 7.33/2.77 U33_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> FRONTA_IN_AA(X5, X10) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, .(X7, X8))) -> U35_AG(X1, X2, X3, X4, X5, X6, X7, X8, frontcA_in_aa(X3, X9)) 7.33/2.77 U35_AG(X1, X2, X3, X4, X5, X6, X7, X8, frontcA_out_aa(X3, X9)) -> U36_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, frontcA_in_aa(X4, X10)) 7.33/2.77 U36_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, frontcA_out_aa(X4, X10)) -> U37_AG(X1, X2, X3, X4, X5, X6, X7, X8, appcB_in_gga(X9, X10, .(X6, .(X7, X11)))) 7.33/2.77 U37_AG(X1, X2, X3, X4, X5, X6, X7, X8, appcB_out_gga(X9, X10, .(X6, .(X7, X11)))) -> U38_AG(X1, X2, X3, X4, X5, X6, X7, X8, X11, frontcA_in_aa(X5, X12)) 7.33/2.77 U38_AG(X1, X2, X3, X4, X5, X6, X7, X8, X11, frontcA_out_aa(X5, X12)) -> U39_AG(X1, X2, X3, X4, X5, X6, X7, X8, appE_in_ggg(X11, X12, X8)) 7.33/2.77 U38_AG(X1, X2, X3, X4, X5, X6, X7, X8, X11, frontcA_out_aa(X5, X12)) -> APPE_IN_GGG(X11, X12, X8) 7.33/2.77 APPE_IN_GGG(.(X1, X2), X3, .(X1, X4)) -> U7_GGG(X1, X2, X3, X4, appE_in_ggg(X2, X3, X4)) 7.33/2.77 APPE_IN_GGG(.(X1, X2), X3, .(X1, X4)) -> APPE_IN_GGG(X2, X3, X4) 7.33/2.77 7.33/2.77 The TRS R consists of the following rules: 7.33/2.77 7.33/2.77 frontcA_in_aa(void, []) -> frontcA_out_aa(void, []) 7.33/2.77 frontcA_in_aa(tree(X1, void, void), .(X1, [])) -> frontcA_out_aa(tree(X1, void, void), .(X1, [])) 7.33/2.77 frontcA_in_aa(tree(X1, X2, X3), X4) -> U41_aa(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U41_aa(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U42_aa(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U42_aa(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U43_aa(X1, X2, X3, X4, appcB_in_gga(X5, X6, X4)) 7.33/2.77 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 7.33/2.77 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U44_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 7.33/2.77 U44_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 7.33/2.77 U43_aa(X1, X2, X3, X4, appcB_out_gga(X5, X6, X4)) -> frontcA_out_aa(tree(X1, X2, X3), X4) 7.33/2.77 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 [] = [] 7.33/2.77 7.33/2.77 frontA_in_aa(x1, x2) = frontA_in_aa 7.33/2.77 7.33/2.77 frontcA_in_aa(x1, x2) = frontcA_in_aa 7.33/2.77 7.33/2.77 frontcA_out_aa(x1, x2) = frontcA_out_aa(x1, x2) 7.33/2.77 7.33/2.77 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.77 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 U41_aa(x1, x2, x3, x4, x5) = U41_aa(x5) 7.33/2.77 7.33/2.77 U42_aa(x1, x2, x3, x4, x5, x6) = U42_aa(x2, x5, x6) 7.33/2.77 7.33/2.77 U43_aa(x1, x2, x3, x4, x5) = U43_aa(x2, x3, x5) 7.33/2.77 7.33/2.77 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 7.33/2.77 7.33/2.77 U44_gga(x1, x2, x3, x4, x5) = U44_gga(x2, x3, x5) 7.33/2.77 7.33/2.77 appB_in_gga(x1, x2, x3) = appB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appE_in_ggg(x1, x2, x3) = appE_in_ggg(x1, x2, x3) 7.33/2.77 7.33/2.77 FRONTF_IN_AG(x1, x2) = FRONTF_IN_AG(x2) 7.33/2.77 7.33/2.77 U8_AG(x1, x2, x3) = U8_AG(x3) 7.33/2.77 7.33/2.77 FRONTA_IN_AA(x1, x2) = FRONTA_IN_AA 7.33/2.77 7.33/2.77 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.77 7.33/2.77 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 7.33/2.77 7.33/2.77 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 7.33/2.77 7.33/2.77 U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x5, x6) 7.33/2.77 7.33/2.77 U5_AA(x1, x2, x3, x4, x5) = U5_AA(x5) 7.33/2.77 7.33/2.77 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 7.33/2.77 7.33/2.77 U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x2, x3, x5) 7.33/2.77 7.33/2.77 U9_AG(x1, x2, x3, x4) = U9_AG(x4) 7.33/2.77 7.33/2.77 U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x6) 7.33/2.77 7.33/2.77 U11_AG(x1, x2, x3, x4, x5, x6) = U11_AG(x6) 7.33/2.77 7.33/2.77 U12_AG(x1, x2, x3, x4, x5, x6) = U12_AG(x6) 7.33/2.77 7.33/2.77 U13_AG(x1, x2, x3, x4, x5, x6, x7) = U13_AG(x6, x7) 7.33/2.77 7.33/2.77 U14_AG(x1, x2, x3, x4, x5, x6) = U14_AG(x6) 7.33/2.77 7.33/2.77 U15_AG(x1, x2, x3, x4, x5, x6) = U15_AG(x6) 7.33/2.77 7.33/2.77 U16_AG(x1, x2, x3, x4, x5, x6) = U16_AG(x6) 7.33/2.77 7.33/2.77 U17_AG(x1, x2, x3, x4) = U17_AG(x4) 7.33/2.77 7.33/2.77 U18_AG(x1, x2, x3, x4, x5) = U18_AG(x5) 7.33/2.77 7.33/2.77 U19_AG(x1, x2, x3, x4, x5, x6, x7) = U19_AG(x7) 7.33/2.77 7.33/2.77 U20_AG(x1, x2, x3, x4, x5, x6, x7) = U20_AG(x7) 7.33/2.77 7.33/2.77 U21_AG(x1, x2, x3, x4, x5, x6, x7) = U21_AG(x7) 7.33/2.77 7.33/2.77 U22_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U22_AG(x7, x8) 7.33/2.77 7.33/2.77 U23_AG(x1, x2, x3, x4, x5, x6, x7) = U23_AG(x7) 7.33/2.77 7.33/2.77 U24_AG(x1, x2, x3, x4, x5, x6, x7) = U24_AG(x7) 7.33/2.77 7.33/2.77 U25_AG(x1, x2, x3, x4, x5, x6, x7) = U25_AG(x7) 7.33/2.77 7.33/2.77 U26_AG(x1, x2, x3, x4) = U26_AG(x3, x4) 7.33/2.77 7.33/2.77 U27_AG(x1, x2, x3, x4, x5) = U27_AG(x4, x5) 7.33/2.77 7.33/2.77 U28_AG(x1, x2, x3, x4, x5, x6, x7) = U28_AG(x6, x7) 7.33/2.77 7.33/2.77 U29_AG(x1, x2, x3, x4, x5, x6, x7) = U29_AG(x6, x7) 7.33/2.77 7.33/2.77 U30_AG(x1, x2, x3, x4, x5, x6, x7) = U30_AG(x6, x7) 7.33/2.77 7.33/2.77 U31_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U31_AG(x6, x7, x8) 7.33/2.77 7.33/2.77 U32_AG(x1, x2, x3, x4, x5, x6, x7) = U32_AG(x6, x7) 7.33/2.77 7.33/2.77 U33_AG(x1, x2, x3, x4, x5, x6, x7) = U33_AG(x6, x7) 7.33/2.77 7.33/2.77 U34_AG(x1, x2, x3, x4, x5, x6, x7) = U34_AG(x6, x7) 7.33/2.77 7.33/2.77 U35_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_AG(x8, x9) 7.33/2.77 7.33/2.77 U36_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U36_AG(x8, x9, x10) 7.33/2.77 7.33/2.77 U37_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U37_AG(x8, x9) 7.33/2.77 7.33/2.77 U38_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U38_AG(x8, x9, x10) 7.33/2.77 7.33/2.77 U39_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U39_AG(x8, x9) 7.33/2.77 7.33/2.77 APPE_IN_GGG(x1, x2, x3) = APPE_IN_GGG(x1, x2, x3) 7.33/2.77 7.33/2.77 U7_GGG(x1, x2, x3, x4, x5) = U7_GGG(x2, x3, x4, x5) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 7.33/2.77 7.33/2.77 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 7.33/2.77 7.33/2.77 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (66) 7.33/2.77 Obligation: 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), []) -> U8_AG(X1, X2, frontA_in_aa(X2, X3)) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), []) -> FRONTA_IN_AA(X2, X3) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> U1_AA(X1, X2, X3, X4, frontA_in_aa(X2, X5)) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> FRONTA_IN_AA(X2, X5) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> U2_AA(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U3_AA(X1, X2, X3, X4, frontA_in_aa(X3, X6)) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> FRONTA_IN_AA(X3, X6) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U4_AA(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U4_AA(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U5_AA(X1, X2, X3, X4, appB_in_gga(X5, X6, X4)) 7.33/2.77 U4_AA(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> APPB_IN_GGA(X5, X6, X4) 7.33/2.77 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> U6_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4)) 7.33/2.77 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), []) -> U9_AG(X1, X2, X3, frontA_in_aa(X3, X4)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), []) -> FRONTA_IN_AA(X3, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), []) -> U10_AG(X1, X2, X3, X4, X5, frontA_in_aa(X3, X6)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), []) -> FRONTA_IN_AA(X3, X6) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), []) -> U11_AG(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U11_AG(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U12_AG(X1, X2, X3, X4, X5, frontA_in_aa(X4, X7)) 7.33/2.77 U11_AG(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> FRONTA_IN_AA(X4, X7) 7.33/2.77 U11_AG(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U13_AG(X1, X2, X3, X4, X5, X6, frontcA_in_aa(X4, X7)) 7.33/2.77 U13_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X4, X7)) -> U14_AG(X1, X2, X3, X4, X5, appB_in_gga(X6, X7, X8)) 7.33/2.77 U13_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X4, X7)) -> APPB_IN_GGA(X6, X7, X8) 7.33/2.77 U13_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X4, X7)) -> U15_AG(X1, X2, X3, X4, X5, appcB_in_gga(X6, X7, X8)) 7.33/2.77 U15_AG(X1, X2, X3, X4, X5, appcB_out_gga(X6, X7, X8)) -> U16_AG(X1, X2, X3, X4, X5, frontA_in_aa(X5, X9)) 7.33/2.77 U15_AG(X1, X2, X3, X4, X5, appcB_out_gga(X6, X7, X8)) -> FRONTA_IN_AA(X5, X9) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), .(X3, [])) -> U17_AG(X1, X2, X3, frontA_in_aa(X2, X4)) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), .(X3, [])) -> FRONTA_IN_AA(X2, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), .(X4, [])) -> U18_AG(X1, X2, X3, X4, frontA_in_aa(X3, X5)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), .(X4, [])) -> FRONTA_IN_AA(X3, X5) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) -> U19_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X3, X7)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) -> FRONTA_IN_AA(X3, X7) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, [])) -> U20_AG(X1, X2, X3, X4, X5, X6, frontcA_in_aa(X3, X7)) 7.33/2.77 U20_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U21_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X4, X8)) 7.33/2.77 U20_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> FRONTA_IN_AA(X4, X8) 7.33/2.77 U20_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_in_aa(X4, X8)) 7.33/2.77 U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U23_AG(X1, X2, X3, X4, X5, X6, appB_in_gga(X7, X8, X9)) 7.33/2.77 U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> APPB_IN_GGA(X7, X8, X9) 7.33/2.77 U22_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U24_AG(X1, X2, X3, X4, X5, X6, appcB_in_gga(X7, X8, X9)) 7.33/2.77 U24_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> U25_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X5, X10)) 7.33/2.77 U24_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> FRONTA_IN_AA(X5, X10) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), X3) -> U26_AG(X1, X2, X3, frontA_in_aa(X2, X4)) 7.33/2.77 FRONTF_IN_AG(tree(X1, void, X2), X3) -> FRONTA_IN_AA(X2, X4) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), X4) -> U27_AG(X1, X2, X3, X4, frontA_in_aa(X3, X5)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, void, void), X3), X4) -> FRONTA_IN_AA(X3, X5) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U28_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X3, X7)) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> FRONTA_IN_AA(X3, X7) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U29_AG(X1, X2, X3, X4, X5, X6, frontcA_in_aa(X3, X7)) 7.33/2.77 U29_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U30_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X4, X8)) 7.33/2.77 U29_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> FRONTA_IN_AA(X4, X8) 7.33/2.77 U29_AG(X1, X2, X3, X4, X5, X6, frontcA_out_aa(X3, X7)) -> U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_in_aa(X4, X8)) 7.33/2.77 U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U32_AG(X1, X2, X3, X4, X5, X6, appB_in_gga(X7, X8, X9)) 7.33/2.77 U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> APPB_IN_GGA(X7, X8, X9) 7.33/2.77 U31_AG(X1, X2, X3, X4, X5, X6, X7, frontcA_out_aa(X4, X8)) -> U33_AG(X1, X2, X3, X4, X5, X6, appcB_in_gga(X7, X8, X9)) 7.33/2.77 U33_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> U34_AG(X1, X2, X3, X4, X5, X6, frontA_in_aa(X5, X10)) 7.33/2.77 U33_AG(X1, X2, X3, X4, X5, X6, appcB_out_gga(X7, X8, X9)) -> FRONTA_IN_AA(X5, X10) 7.33/2.77 FRONTF_IN_AG(tree(X1, tree(X2, X3, X4), X5), .(X6, .(X7, X8))) -> U35_AG(X1, X2, X3, X4, X5, X6, X7, X8, frontcA_in_aa(X3, X9)) 7.33/2.77 U35_AG(X1, X2, X3, X4, X5, X6, X7, X8, frontcA_out_aa(X3, X9)) -> U36_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, frontcA_in_aa(X4, X10)) 7.33/2.77 U36_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, frontcA_out_aa(X4, X10)) -> U37_AG(X1, X2, X3, X4, X5, X6, X7, X8, appcB_in_gga(X9, X10, .(X6, .(X7, X11)))) 7.33/2.77 U37_AG(X1, X2, X3, X4, X5, X6, X7, X8, appcB_out_gga(X9, X10, .(X6, .(X7, X11)))) -> U38_AG(X1, X2, X3, X4, X5, X6, X7, X8, X11, frontcA_in_aa(X5, X12)) 7.33/2.77 U38_AG(X1, X2, X3, X4, X5, X6, X7, X8, X11, frontcA_out_aa(X5, X12)) -> U39_AG(X1, X2, X3, X4, X5, X6, X7, X8, appE_in_ggg(X11, X12, X8)) 7.33/2.77 U38_AG(X1, X2, X3, X4, X5, X6, X7, X8, X11, frontcA_out_aa(X5, X12)) -> APPE_IN_GGG(X11, X12, X8) 7.33/2.77 APPE_IN_GGG(.(X1, X2), X3, .(X1, X4)) -> U7_GGG(X1, X2, X3, X4, appE_in_ggg(X2, X3, X4)) 7.33/2.77 APPE_IN_GGG(.(X1, X2), X3, .(X1, X4)) -> APPE_IN_GGG(X2, X3, X4) 7.33/2.77 7.33/2.77 The TRS R consists of the following rules: 7.33/2.77 7.33/2.77 frontcA_in_aa(void, []) -> frontcA_out_aa(void, []) 7.33/2.77 frontcA_in_aa(tree(X1, void, void), .(X1, [])) -> frontcA_out_aa(tree(X1, void, void), .(X1, [])) 7.33/2.77 frontcA_in_aa(tree(X1, X2, X3), X4) -> U41_aa(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U41_aa(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U42_aa(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U42_aa(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U43_aa(X1, X2, X3, X4, appcB_in_gga(X5, X6, X4)) 7.33/2.77 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 7.33/2.77 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U44_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 7.33/2.77 U44_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 7.33/2.77 U43_aa(X1, X2, X3, X4, appcB_out_gga(X5, X6, X4)) -> frontcA_out_aa(tree(X1, X2, X3), X4) 7.33/2.77 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 [] = [] 7.33/2.77 7.33/2.77 frontA_in_aa(x1, x2) = frontA_in_aa 7.33/2.77 7.33/2.77 frontcA_in_aa(x1, x2) = frontcA_in_aa 7.33/2.77 7.33/2.77 frontcA_out_aa(x1, x2) = frontcA_out_aa(x1, x2) 7.33/2.77 7.33/2.77 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.77 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 U41_aa(x1, x2, x3, x4, x5) = U41_aa(x5) 7.33/2.77 7.33/2.77 U42_aa(x1, x2, x3, x4, x5, x6) = U42_aa(x2, x5, x6) 7.33/2.77 7.33/2.77 U43_aa(x1, x2, x3, x4, x5) = U43_aa(x2, x3, x5) 7.33/2.77 7.33/2.77 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 7.33/2.77 7.33/2.77 U44_gga(x1, x2, x3, x4, x5) = U44_gga(x2, x3, x5) 7.33/2.77 7.33/2.77 appB_in_gga(x1, x2, x3) = appB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appE_in_ggg(x1, x2, x3) = appE_in_ggg(x1, x2, x3) 7.33/2.77 7.33/2.77 FRONTF_IN_AG(x1, x2) = FRONTF_IN_AG(x2) 7.33/2.77 7.33/2.77 U8_AG(x1, x2, x3) = U8_AG(x3) 7.33/2.77 7.33/2.77 FRONTA_IN_AA(x1, x2) = FRONTA_IN_AA 7.33/2.77 7.33/2.77 U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) 7.33/2.77 7.33/2.77 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 7.33/2.77 7.33/2.77 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 7.33/2.77 7.33/2.77 U4_AA(x1, x2, x3, x4, x5, x6) = U4_AA(x5, x6) 7.33/2.77 7.33/2.77 U5_AA(x1, x2, x3, x4, x5) = U5_AA(x5) 7.33/2.77 7.33/2.77 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 7.33/2.77 7.33/2.77 U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x2, x3, x5) 7.33/2.77 7.33/2.77 U9_AG(x1, x2, x3, x4) = U9_AG(x4) 7.33/2.77 7.33/2.77 U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x6) 7.33/2.77 7.33/2.77 U11_AG(x1, x2, x3, x4, x5, x6) = U11_AG(x6) 7.33/2.77 7.33/2.77 U12_AG(x1, x2, x3, x4, x5, x6) = U12_AG(x6) 7.33/2.77 7.33/2.77 U13_AG(x1, x2, x3, x4, x5, x6, x7) = U13_AG(x6, x7) 7.33/2.77 7.33/2.77 U14_AG(x1, x2, x3, x4, x5, x6) = U14_AG(x6) 7.33/2.77 7.33/2.77 U15_AG(x1, x2, x3, x4, x5, x6) = U15_AG(x6) 7.33/2.77 7.33/2.77 U16_AG(x1, x2, x3, x4, x5, x6) = U16_AG(x6) 7.33/2.77 7.33/2.77 U17_AG(x1, x2, x3, x4) = U17_AG(x4) 7.33/2.77 7.33/2.77 U18_AG(x1, x2, x3, x4, x5) = U18_AG(x5) 7.33/2.77 7.33/2.77 U19_AG(x1, x2, x3, x4, x5, x6, x7) = U19_AG(x7) 7.33/2.77 7.33/2.77 U20_AG(x1, x2, x3, x4, x5, x6, x7) = U20_AG(x7) 7.33/2.77 7.33/2.77 U21_AG(x1, x2, x3, x4, x5, x6, x7) = U21_AG(x7) 7.33/2.77 7.33/2.77 U22_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U22_AG(x7, x8) 7.33/2.77 7.33/2.77 U23_AG(x1, x2, x3, x4, x5, x6, x7) = U23_AG(x7) 7.33/2.77 7.33/2.77 U24_AG(x1, x2, x3, x4, x5, x6, x7) = U24_AG(x7) 7.33/2.77 7.33/2.77 U25_AG(x1, x2, x3, x4, x5, x6, x7) = U25_AG(x7) 7.33/2.77 7.33/2.77 U26_AG(x1, x2, x3, x4) = U26_AG(x3, x4) 7.33/2.77 7.33/2.77 U27_AG(x1, x2, x3, x4, x5) = U27_AG(x4, x5) 7.33/2.77 7.33/2.77 U28_AG(x1, x2, x3, x4, x5, x6, x7) = U28_AG(x6, x7) 7.33/2.77 7.33/2.77 U29_AG(x1, x2, x3, x4, x5, x6, x7) = U29_AG(x6, x7) 7.33/2.77 7.33/2.77 U30_AG(x1, x2, x3, x4, x5, x6, x7) = U30_AG(x6, x7) 7.33/2.77 7.33/2.77 U31_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U31_AG(x6, x7, x8) 7.33/2.77 7.33/2.77 U32_AG(x1, x2, x3, x4, x5, x6, x7) = U32_AG(x6, x7) 7.33/2.77 7.33/2.77 U33_AG(x1, x2, x3, x4, x5, x6, x7) = U33_AG(x6, x7) 7.33/2.77 7.33/2.77 U34_AG(x1, x2, x3, x4, x5, x6, x7) = U34_AG(x6, x7) 7.33/2.77 7.33/2.77 U35_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_AG(x8, x9) 7.33/2.77 7.33/2.77 U36_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U36_AG(x8, x9, x10) 7.33/2.77 7.33/2.77 U37_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U37_AG(x8, x9) 7.33/2.77 7.33/2.77 U38_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U38_AG(x8, x9, x10) 7.33/2.77 7.33/2.77 U39_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U39_AG(x8, x9) 7.33/2.77 7.33/2.77 APPE_IN_GGG(x1, x2, x3) = APPE_IN_GGG(x1, x2, x3) 7.33/2.77 7.33/2.77 U7_GGG(x1, x2, x3, x4, x5) = U7_GGG(x2, x3, x4, x5) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (67) DependencyGraphProof (EQUIVALENT) 7.33/2.77 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 58 less nodes. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (68) 7.33/2.77 Complex Obligation (AND) 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (69) 7.33/2.77 Obligation: 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 APPE_IN_GGG(.(X1, X2), X3, .(X1, X4)) -> APPE_IN_GGG(X2, X3, X4) 7.33/2.77 7.33/2.77 The TRS R consists of the following rules: 7.33/2.77 7.33/2.77 frontcA_in_aa(void, []) -> frontcA_out_aa(void, []) 7.33/2.77 frontcA_in_aa(tree(X1, void, void), .(X1, [])) -> frontcA_out_aa(tree(X1, void, void), .(X1, [])) 7.33/2.77 frontcA_in_aa(tree(X1, X2, X3), X4) -> U41_aa(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U41_aa(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U42_aa(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U42_aa(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U43_aa(X1, X2, X3, X4, appcB_in_gga(X5, X6, X4)) 7.33/2.77 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 7.33/2.77 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U44_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 7.33/2.77 U44_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 7.33/2.77 U43_aa(X1, X2, X3, X4, appcB_out_gga(X5, X6, X4)) -> frontcA_out_aa(tree(X1, X2, X3), X4) 7.33/2.77 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 [] = [] 7.33/2.77 7.33/2.77 frontcA_in_aa(x1, x2) = frontcA_in_aa 7.33/2.77 7.33/2.77 frontcA_out_aa(x1, x2) = frontcA_out_aa(x1, x2) 7.33/2.77 7.33/2.77 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.77 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 U41_aa(x1, x2, x3, x4, x5) = U41_aa(x5) 7.33/2.77 7.33/2.77 U42_aa(x1, x2, x3, x4, x5, x6) = U42_aa(x2, x5, x6) 7.33/2.77 7.33/2.77 U43_aa(x1, x2, x3, x4, x5) = U43_aa(x2, x3, x5) 7.33/2.77 7.33/2.77 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 7.33/2.77 7.33/2.77 U44_gga(x1, x2, x3, x4, x5) = U44_gga(x2, x3, x5) 7.33/2.77 7.33/2.77 APPE_IN_GGG(x1, x2, x3) = APPE_IN_GGG(x1, x2, x3) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (70) UsableRulesProof (EQUIVALENT) 7.33/2.77 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (71) 7.33/2.77 Obligation: 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 APPE_IN_GGG(.(X1, X2), X3, .(X1, X4)) -> APPE_IN_GGG(X2, X3, X4) 7.33/2.77 7.33/2.77 R is empty. 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 APPE_IN_GGG(x1, x2, x3) = APPE_IN_GGG(x1, x2, x3) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (72) PiDPToQDPProof (SOUND) 7.33/2.77 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (73) 7.33/2.77 Obligation: 7.33/2.77 Q DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 APPE_IN_GGG(.(X2), X3, .(X4)) -> APPE_IN_GGG(X2, X3, X4) 7.33/2.77 7.33/2.77 R is empty. 7.33/2.77 Q is empty. 7.33/2.77 We have to consider all (P,Q,R)-chains. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (74) QDPSizeChangeProof (EQUIVALENT) 7.33/2.77 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. 7.33/2.77 7.33/2.77 From the DPs we obtained the following set of size-change graphs: 7.33/2.77 *APPE_IN_GGG(.(X2), X3, .(X4)) -> APPE_IN_GGG(X2, X3, X4) 7.33/2.77 The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3 7.33/2.77 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (75) 7.33/2.77 YES 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (76) 7.33/2.77 Obligation: 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 7.33/2.77 7.33/2.77 The TRS R consists of the following rules: 7.33/2.77 7.33/2.77 frontcA_in_aa(void, []) -> frontcA_out_aa(void, []) 7.33/2.77 frontcA_in_aa(tree(X1, void, void), .(X1, [])) -> frontcA_out_aa(tree(X1, void, void), .(X1, [])) 7.33/2.77 frontcA_in_aa(tree(X1, X2, X3), X4) -> U41_aa(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U41_aa(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U42_aa(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U42_aa(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U43_aa(X1, X2, X3, X4, appcB_in_gga(X5, X6, X4)) 7.33/2.77 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 7.33/2.77 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U44_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 7.33/2.77 U44_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 7.33/2.77 U43_aa(X1, X2, X3, X4, appcB_out_gga(X5, X6, X4)) -> frontcA_out_aa(tree(X1, X2, X3), X4) 7.33/2.77 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 [] = [] 7.33/2.77 7.33/2.77 frontcA_in_aa(x1, x2) = frontcA_in_aa 7.33/2.77 7.33/2.77 frontcA_out_aa(x1, x2) = frontcA_out_aa(x1, x2) 7.33/2.77 7.33/2.77 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.77 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 U41_aa(x1, x2, x3, x4, x5) = U41_aa(x5) 7.33/2.77 7.33/2.77 U42_aa(x1, x2, x3, x4, x5, x6) = U42_aa(x2, x5, x6) 7.33/2.77 7.33/2.77 U43_aa(x1, x2, x3, x4, x5) = U43_aa(x2, x3, x5) 7.33/2.77 7.33/2.77 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 7.33/2.77 7.33/2.77 U44_gga(x1, x2, x3, x4, x5) = U44_gga(x2, x3, x5) 7.33/2.77 7.33/2.77 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (77) UsableRulesProof (EQUIVALENT) 7.33/2.77 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (78) 7.33/2.77 Obligation: 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 7.33/2.77 7.33/2.77 R is empty. 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (79) PiDPToQDPProof (SOUND) 7.33/2.77 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (80) 7.33/2.77 Obligation: 7.33/2.77 Q DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 APPB_IN_GGA(.(X2), X3) -> APPB_IN_GGA(X2, X3) 7.33/2.77 7.33/2.77 R is empty. 7.33/2.77 Q is empty. 7.33/2.77 We have to consider all (P,Q,R)-chains. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (81) QDPSizeChangeProof (EQUIVALENT) 7.33/2.77 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. 7.33/2.77 7.33/2.77 From the DPs we obtained the following set of size-change graphs: 7.33/2.77 *APPB_IN_GGA(.(X2), X3) -> APPB_IN_GGA(X2, X3) 7.33/2.77 The graph contains the following edges 1 > 1, 2 >= 2 7.33/2.77 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (82) 7.33/2.77 YES 7.33/2.77 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (83) 7.33/2.77 Obligation: 7.33/2.77 Pi DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> U2_AA(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U2_AA(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> FRONTA_IN_AA(X3, X6) 7.33/2.77 FRONTA_IN_AA(tree(X1, X2, X3), X4) -> FRONTA_IN_AA(X2, X5) 7.33/2.77 7.33/2.77 The TRS R consists of the following rules: 7.33/2.77 7.33/2.77 frontcA_in_aa(void, []) -> frontcA_out_aa(void, []) 7.33/2.77 frontcA_in_aa(tree(X1, void, void), .(X1, [])) -> frontcA_out_aa(tree(X1, void, void), .(X1, [])) 7.33/2.77 frontcA_in_aa(tree(X1, X2, X3), X4) -> U41_aa(X1, X2, X3, X4, frontcA_in_aa(X2, X5)) 7.33/2.77 U41_aa(X1, X2, X3, X4, frontcA_out_aa(X2, X5)) -> U42_aa(X1, X2, X3, X4, X5, frontcA_in_aa(X3, X6)) 7.33/2.77 U42_aa(X1, X2, X3, X4, X5, frontcA_out_aa(X3, X6)) -> U43_aa(X1, X2, X3, X4, appcB_in_gga(X5, X6, X4)) 7.33/2.77 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 7.33/2.77 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U44_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 7.33/2.77 U44_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 7.33/2.77 U43_aa(X1, X2, X3, X4, appcB_out_gga(X5, X6, X4)) -> frontcA_out_aa(tree(X1, X2, X3), X4) 7.33/2.77 7.33/2.77 The argument filtering Pi contains the following mapping: 7.33/2.77 [] = [] 7.33/2.77 7.33/2.77 frontcA_in_aa(x1, x2) = frontcA_in_aa 7.33/2.77 7.33/2.77 frontcA_out_aa(x1, x2) = frontcA_out_aa(x1, x2) 7.33/2.77 7.33/2.77 tree(x1, x2, x3) = tree(x2, x3) 7.33/2.77 7.33/2.77 .(x1, x2) = .(x2) 7.33/2.77 7.33/2.77 U41_aa(x1, x2, x3, x4, x5) = U41_aa(x5) 7.33/2.77 7.33/2.77 U42_aa(x1, x2, x3, x4, x5, x6) = U42_aa(x2, x5, x6) 7.33/2.77 7.33/2.77 U43_aa(x1, x2, x3, x4, x5) = U43_aa(x2, x3, x5) 7.33/2.77 7.33/2.77 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 7.33/2.77 7.33/2.77 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 7.33/2.77 7.33/2.77 U44_gga(x1, x2, x3, x4, x5) = U44_gga(x2, x3, x5) 7.33/2.77 7.33/2.77 FRONTA_IN_AA(x1, x2) = FRONTA_IN_AA 7.33/2.77 7.33/2.77 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 7.33/2.77 7.33/2.77 7.33/2.77 We have to consider all (P,R,Pi)-chains 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (84) PiDPToQDPProof (SOUND) 7.33/2.77 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (85) 7.33/2.77 Obligation: 7.33/2.77 Q DP problem: 7.33/2.77 The TRS P consists of the following rules: 7.33/2.77 7.33/2.77 FRONTA_IN_AA -> U2_AA(frontcA_in_aa) 7.33/2.77 U2_AA(frontcA_out_aa(X2, X5)) -> FRONTA_IN_AA 7.33/2.77 FRONTA_IN_AA -> FRONTA_IN_AA 7.33/2.77 7.33/2.77 The TRS R consists of the following rules: 7.33/2.77 7.33/2.77 frontcA_in_aa -> frontcA_out_aa(void, []) 7.33/2.77 frontcA_in_aa -> frontcA_out_aa(tree(void, void), .([])) 7.33/2.77 frontcA_in_aa -> U41_aa(frontcA_in_aa) 7.33/2.77 U41_aa(frontcA_out_aa(X2, X5)) -> U42_aa(X2, X5, frontcA_in_aa) 7.33/2.77 U42_aa(X2, X5, frontcA_out_aa(X3, X6)) -> U43_aa(X2, X3, appcB_in_gga(X5, X6)) 7.33/2.77 appcB_in_gga([], X1) -> appcB_out_gga([], X1, X1) 7.33/2.77 appcB_in_gga(.(X2), X3) -> U44_gga(X2, X3, appcB_in_gga(X2, X3)) 7.33/2.77 U44_gga(X2, X3, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X2), X3, .(X4)) 7.33/2.77 U43_aa(X2, X3, appcB_out_gga(X5, X6, X4)) -> frontcA_out_aa(tree(X2, X3), X4) 7.33/2.77 7.33/2.77 The set Q consists of the following terms: 7.33/2.77 7.33/2.77 frontcA_in_aa 7.33/2.77 U41_aa(x0) 7.33/2.77 U42_aa(x0, x1, x2) 7.33/2.77 appcB_in_gga(x0, x1) 7.33/2.77 U44_gga(x0, x1, x2) 7.33/2.77 U43_aa(x0, x1, x2) 7.33/2.77 7.33/2.77 We have to consider all (P,Q,R)-chains. 7.33/2.77 ---------------------------------------- 7.33/2.77 7.33/2.77 (86) PrologToIRSwTTransformerProof (SOUND) 7.33/2.77 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 7.33/2.77 7.33/2.77 { 7.33/2.77 "root": 211, 7.33/2.77 "program": { 7.33/2.77 "directives": [], 7.33/2.77 "clauses": [ 7.33/2.77 [ 7.33/2.77 "(front (void) ([]))", 7.33/2.77 null 7.33/2.77 ], 7.33/2.77 [ 7.33/2.77 "(front (tree X (void) (void)) (. X ([])))", 7.33/2.77 null 7.33/2.77 ], 7.33/2.77 [ 7.33/2.77 "(front (tree X1 L R) Xs)", 7.33/2.77 "(',' (front L Ls) (',' (front R Rs) (app Ls Rs Xs)))" 7.33/2.77 ], 7.33/2.77 [ 7.33/2.77 "(app ([]) X X)", 7.33/2.77 null 7.33/2.77 ], 7.33/2.77 [ 7.33/2.77 "(app (. X Xs) Ys (. 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"intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": ["X46"], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "220": { 7.33/2.77 "goal": [], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "264": { 7.33/2.77 "goal": [{ 7.33/2.77 "clause": -1, 7.33/2.77 "scope": -1, 7.33/2.77 "term": "(',' (front T23 X22) (app T22 X22 T19))" 7.33/2.77 }], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": ["T19"], 7.33/2.77 "free": ["X22"], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "341": { 7.33/2.77 "goal": [{ 7.33/2.77 "clause": -1, 7.33/2.77 "scope": -1, 7.33/2.77 "term": "(true)" 7.33/2.77 }], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "342": { 7.33/2.77 "goal": [], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "343": { 7.33/2.77 "goal": [], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "223": { 7.33/2.77 "goal": [{ 7.33/2.77 "clause": 1, 7.33/2.77 "scope": 1, 7.33/2.77 "term": "(front T1 T2)" 7.33/2.77 }], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": ["T2"], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "267": { 7.33/2.77 "goal": [ 7.33/2.77 { 7.33/2.77 "clause": 0, 7.33/2.77 "scope": 2, 7.33/2.77 "term": "(front T20 X21)" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "clause": 1, 7.33/2.77 "scope": 2, 7.33/2.77 "term": "(front T20 X21)" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "clause": 2, 7.33/2.77 "scope": 2, 7.33/2.77 "term": "(front T20 X21)" 7.33/2.77 } 7.33/2.77 ], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": ["X21"], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "344": { 7.33/2.77 "goal": [{ 7.33/2.77 "clause": -1, 7.33/2.77 "scope": -1, 7.33/2.77 "term": "(app T60 T61 X68)" 7.33/2.77 }], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": ["X68"], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "268": { 7.33/2.77 "goal": [{ 7.33/2.77 "clause": 0, 7.33/2.77 "scope": 2, 7.33/2.77 "term": "(front T20 X21)" 7.33/2.77 }], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": ["X21"], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "345": { 7.33/2.77 "goal": [], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "225": { 7.33/2.77 "goal": [{ 7.33/2.77 "clause": 2, 7.33/2.77 "scope": 1, 7.33/2.77 "term": "(front T1 T2)" 7.33/2.77 }], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": ["T2"], 7.33/2.77 "free": [], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "269": { 7.33/2.77 "goal": [ 7.33/2.77 { 7.33/2.77 "clause": 1, 7.33/2.77 "scope": 2, 7.33/2.77 "term": "(front T20 X21)" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "clause": 2, 7.33/2.77 "scope": 2, 7.33/2.77 "term": "(front T20 X21)" 7.33/2.77 } 7.33/2.77 ], 7.33/2.77 "kb": { 7.33/2.77 "nonunifying": [], 7.33/2.77 "intvars": {}, 7.33/2.77 "arithmetic": { 7.33/2.77 "type": "PlainIntegerRelationState", 7.33/2.77 "relations": [] 7.33/2.77 }, 7.33/2.77 "ground": [], 7.33/2.77 "free": ["X21"], 7.33/2.77 "exprvars": [] 7.33/2.77 } 7.33/2.77 } 7.33/2.77 }, 7.33/2.77 "edges": [ 7.33/2.77 { 7.33/2.77 "from": 211, 7.33/2.77 "to": 215, 7.33/2.77 "label": "CASE" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 215, 7.33/2.77 "to": 216, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 215, 7.33/2.77 "to": 217, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 216, 7.33/2.77 "to": 218, 7.33/2.77 "label": "EVAL with clause\nfront(void, []).\nand substitutionT1 -> void,\nT2 -> []" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 216, 7.33/2.77 "to": 219, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 217, 7.33/2.77 "to": 223, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 217, 7.33/2.77 "to": 225, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 218, 7.33/2.77 "to": 220, 7.33/2.77 "label": "SUCCESS" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 223, 7.33/2.77 "to": 230, 7.33/2.77 "label": "EVAL with clause\nfront(tree(X6, void, void), .(X6, [])).\nand substitutionX6 -> T7,\nT1 -> tree(T7, void, void),\nT2 -> .(T7, [])" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 223, 7.33/2.77 "to": 233, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 225, 7.33/2.77 "to": 249, 7.33/2.77 "label": "EVAL with clause\nfront(tree(X17, X18, X19), X20) :- ','(front(X18, X21), ','(front(X19, X22), app(X21, X22, X20))).\nand substitutionX17 -> T16,\nX18 -> T20,\nX19 -> T21,\nT1 -> tree(T16, T20, T21),\nT2 -> T19,\nX20 -> T19,\nT17 -> T20,\nT18 -> T21" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 225, 7.33/2.77 "to": 251, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 230, 7.33/2.77 "to": 235, 7.33/2.77 "label": "SUCCESS" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 249, 7.33/2.77 "to": 263, 7.33/2.77 "label": "SPLIT 1" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 249, 7.33/2.77 "to": 264, 7.33/2.77 "label": "SPLIT 2\nreplacements:X21 -> T22,\nT21 -> T23" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 263, 7.33/2.77 "to": 267, 7.33/2.77 "label": "CASE" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 264, 7.33/2.77 "to": 398, 7.33/2.77 "label": "SPLIT 1" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 264, 7.33/2.77 "to": 399, 7.33/2.77 "label": "SPLIT 2\nreplacements:X22 -> T64,\nT22 -> T65" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 267, 7.33/2.77 "to": 268, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 267, 7.33/2.77 "to": 269, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 268, 7.33/2.77 "to": 270, 7.33/2.77 "label": "EVAL with clause\nfront(void, []).\nand substitutionT20 -> void,\nX21 -> []" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 268, 7.33/2.77 "to": 272, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 269, 7.33/2.77 "to": 275, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 269, 7.33/2.77 "to": 277, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 270, 7.33/2.77 "to": 274, 7.33/2.77 "label": "SUCCESS" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 275, 7.33/2.77 "to": 282, 7.33/2.77 "label": "EVAL with clause\nfront(tree(X27, void, void), .(X27, [])).\nand substitutionX27 -> T28,\nT20 -> tree(T28, void, void),\nX21 -> .(T28, [])" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 275, 7.33/2.77 "to": 284, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 277, 7.33/2.77 "to": 291, 7.33/2.77 "label": "EVAL with clause\nfront(tree(X40, X41, X42), X43) :- ','(front(X41, X44), ','(front(X42, X45), app(X44, X45, X43))).\nand substitutionX40 -> T35,\nX41 -> T38,\nX42 -> T39,\nT20 -> tree(T35, T38, T39),\nX21 -> X46,\nX43 -> X46,\nT36 -> T38,\nT37 -> T39" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 277, 7.33/2.77 "to": 292, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 282, 7.33/2.77 "to": 285, 7.33/2.77 "label": "SUCCESS" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 291, 7.33/2.77 "to": 295, 7.33/2.77 "label": "SPLIT 1" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 291, 7.33/2.77 "to": 296, 7.33/2.77 "label": "SPLIT 2\nreplacements:X44 -> T40,\nT39 -> T41" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 295, 7.33/2.77 "to": 263, 7.33/2.77 "label": "INSTANCE with matching:\nT20 -> T38\nX21 -> X44" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 296, 7.33/2.77 "to": 336, 7.33/2.77 "label": "SPLIT 1" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 296, 7.33/2.77 "to": 337, 7.33/2.77 "label": "SPLIT 2\nreplacements:X45 -> T42,\nT40 -> T43" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 336, 7.33/2.77 "to": 263, 7.33/2.77 "label": "INSTANCE with matching:\nT20 -> T41\nX21 -> X45" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 337, 7.33/2.77 "to": 338, 7.33/2.77 "label": "CASE" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 338, 7.33/2.77 "to": 339, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 338, 7.33/2.77 "to": 340, 7.33/2.77 "label": "PARALLEL" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 339, 7.33/2.77 "to": 341, 7.33/2.77 "label": "EVAL with clause\napp([], X53, X53).\nand substitutionT43 -> [],\nT42 -> T50,\nX53 -> T50,\nX46 -> T50" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 339, 7.33/2.77 "to": 342, 7.33/2.77 "label": "EVAL-BACKTRACK" 7.33/2.77 }, 7.33/2.77 { 7.33/2.77 "from": 340, 7.33/2.77 "to": 344, 7.33/2.77 "label": "EVAL with clause\napp(.(X64, X65), X66, .(X64, X67)) :- app(X65, X66, X67).\nand substitutionX64 -> T57,\nX65 -> T60,\nT43 -> .(T57, T60),\nT42 -> T61,\nX66 -> T61,\nX67 -> X68,\nX46 -> .(T57, X68),\nT58 -> T60,\nT59 -> T61" 7.33/2.77 }, 7.33/2.77 { 7.33/2.78 "from": 340, 7.33/2.78 "to": 345, 7.33/2.78 "label": "EVAL-BACKTRACK" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 341, 7.33/2.78 "to": 343, 7.33/2.78 "label": "SUCCESS" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 344, 7.33/2.78 "to": 337, 7.33/2.78 "label": "INSTANCE with matching:\nT43 -> T60\nT42 -> T61\nX46 -> X68" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 398, 7.33/2.78 "to": 263, 7.33/2.78 "label": "INSTANCE with matching:\nT20 -> T23\nX21 -> X22" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 399, 7.33/2.78 "to": 402, 7.33/2.78 "label": "CASE" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 402, 7.33/2.78 "to": 403, 7.33/2.78 "label": "PARALLEL" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 402, 7.33/2.78 "to": 404, 7.33/2.78 "label": "PARALLEL" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 403, 7.33/2.78 "to": 407, 7.33/2.78 "label": "EVAL with clause\napp([], X77, X77).\nand substitutionT65 -> [],\nT64 -> T72,\nX77 -> T72,\nT19 -> T72" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 403, 7.33/2.78 "to": 408, 7.33/2.78 "label": "EVAL-BACKTRACK" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 404, 7.33/2.78 "to": 411, 7.33/2.78 "label": "EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T81,\nX87 -> T85,\nT65 -> .(T81, T85),\nT64 -> T86,\nX88 -> T86,\nX89 -> T84,\nT19 -> .(T81, T84),\nT82 -> T85,\nT83 -> T86" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 404, 7.33/2.78 "to": 412, 7.33/2.78 "label": "EVAL-BACKTRACK" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 407, 7.33/2.78 "to": 410, 7.33/2.78 "label": "SUCCESS" 7.33/2.78 }, 7.33/2.78 { 7.33/2.78 "from": 411, 7.33/2.78 "to": 399, 7.33/2.78 "label": "INSTANCE with matching:\nT65 -> T85\nT64 -> T86\nT19 -> T84" 7.33/2.78 } 7.33/2.78 ], 7.33/2.78 "type": "Graph" 7.33/2.78 } 7.33/2.78 } 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (87) 7.33/2.78 Complex Obligation (AND) 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (88) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f411_in(T84) -> f399_in(T84) :|: TRUE 7.33/2.78 f399_out(x) -> f411_out(x) :|: TRUE 7.33/2.78 f403_out(T19) -> f402_out(T19) :|: TRUE 7.33/2.78 f404_out(x1) -> f402_out(x1) :|: TRUE 7.33/2.78 f402_in(x2) -> f403_in(x2) :|: TRUE 7.33/2.78 f402_in(x3) -> f404_in(x3) :|: TRUE 7.33/2.78 f399_in(x4) -> f402_in(x4) :|: TRUE 7.33/2.78 f402_out(x5) -> f399_out(x5) :|: TRUE 7.33/2.78 f404_in(.(x6, x7)) -> f411_in(x7) :|: TRUE 7.33/2.78 f412_out -> f404_out(x8) :|: TRUE 7.33/2.78 f404_in(x9) -> f412_in :|: TRUE 7.33/2.78 f411_out(x10) -> f404_out(.(x11, x10)) :|: TRUE 7.33/2.78 f211_in(T2) -> f215_in(T2) :|: TRUE 7.33/2.78 f215_out(x12) -> f211_out(x12) :|: TRUE 7.33/2.78 f216_out(x13) -> f215_out(x13) :|: TRUE 7.33/2.78 f215_in(x14) -> f216_in(x14) :|: TRUE 7.33/2.78 f215_in(x15) -> f217_in(x15) :|: TRUE 7.33/2.78 f217_out(x16) -> f215_out(x16) :|: TRUE 7.33/2.78 f223_out(x17) -> f217_out(x17) :|: TRUE 7.33/2.78 f217_in(x18) -> f223_in(x18) :|: TRUE 7.33/2.78 f225_out(x19) -> f217_out(x19) :|: TRUE 7.33/2.78 f217_in(x20) -> f225_in(x20) :|: TRUE 7.33/2.78 f225_in(x21) -> f249_in(x21) :|: TRUE 7.33/2.78 f249_out(x22) -> f225_out(x22) :|: TRUE 7.33/2.78 f225_in(x23) -> f251_in :|: TRUE 7.33/2.78 f251_out -> f225_out(x24) :|: TRUE 7.33/2.78 f263_out -> f264_in(x25) :|: TRUE 7.33/2.78 f264_out(x26) -> f249_out(x26) :|: TRUE 7.33/2.78 f249_in(x27) -> f263_in :|: TRUE 7.33/2.78 f264_in(x28) -> f398_in :|: TRUE 7.33/2.78 f398_out -> f399_in(x29) :|: TRUE 7.33/2.78 f399_out(x30) -> f264_out(x30) :|: TRUE 7.33/2.78 Start term: f211_in(T2) 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (89) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 7.33/2.78 Constructed simple dependency graph. 7.33/2.78 7.33/2.78 Simplified to the following IRSwTs: 7.33/2.78 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (90) 7.33/2.78 TRUE 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (91) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f338_out -> f337_out :|: TRUE 7.33/2.78 f337_in -> f338_in :|: TRUE 7.33/2.78 f345_out -> f340_out :|: TRUE 7.33/2.78 f340_in -> f345_in :|: TRUE 7.33/2.78 f344_out -> f340_out :|: TRUE 7.33/2.78 f340_in -> f344_in :|: TRUE 7.33/2.78 f340_out -> f338_out :|: TRUE 7.33/2.78 f338_in -> f340_in :|: TRUE 7.33/2.78 f339_out -> f338_out :|: TRUE 7.33/2.78 f338_in -> f339_in :|: TRUE 7.33/2.78 f337_out -> f344_out :|: TRUE 7.33/2.78 f344_in -> f337_in :|: TRUE 7.33/2.78 f211_in(T2) -> f215_in(T2) :|: TRUE 7.33/2.78 f215_out(x) -> f211_out(x) :|: TRUE 7.33/2.78 f216_out(x1) -> f215_out(x1) :|: TRUE 7.33/2.78 f215_in(x2) -> f216_in(x2) :|: TRUE 7.33/2.78 f215_in(x3) -> f217_in(x3) :|: TRUE 7.33/2.78 f217_out(x4) -> f215_out(x4) :|: TRUE 7.33/2.78 f223_out(x5) -> f217_out(x5) :|: TRUE 7.33/2.78 f217_in(x6) -> f223_in(x6) :|: TRUE 7.33/2.78 f225_out(x7) -> f217_out(x7) :|: TRUE 7.33/2.78 f217_in(x8) -> f225_in(x8) :|: TRUE 7.33/2.78 f225_in(T19) -> f249_in(T19) :|: TRUE 7.33/2.78 f249_out(x9) -> f225_out(x9) :|: TRUE 7.33/2.78 f225_in(x10) -> f251_in :|: TRUE 7.33/2.78 f251_out -> f225_out(x11) :|: TRUE 7.33/2.78 f263_out -> f264_in(x12) :|: TRUE 7.33/2.78 f264_out(x13) -> f249_out(x13) :|: TRUE 7.33/2.78 f249_in(x14) -> f263_in :|: TRUE 7.33/2.78 f264_in(x15) -> f398_in :|: TRUE 7.33/2.78 f398_out -> f399_in(x16) :|: TRUE 7.33/2.78 f399_out(x17) -> f264_out(x17) :|: TRUE 7.33/2.78 f398_in -> f263_in :|: TRUE 7.33/2.78 f263_out -> f398_out :|: TRUE 7.33/2.78 f267_out -> f263_out :|: TRUE 7.33/2.78 f263_in -> f267_in :|: TRUE 7.33/2.78 f267_in -> f268_in :|: TRUE 7.33/2.78 f268_out -> f267_out :|: TRUE 7.33/2.78 f267_in -> f269_in :|: TRUE 7.33/2.78 f269_out -> f267_out :|: TRUE 7.33/2.78 f277_out -> f269_out :|: TRUE 7.33/2.78 f269_in -> f275_in :|: TRUE 7.33/2.78 f275_out -> f269_out :|: TRUE 7.33/2.78 f269_in -> f277_in :|: TRUE 7.33/2.78 f277_in -> f291_in :|: TRUE 7.33/2.78 f291_out -> f277_out :|: TRUE 7.33/2.78 f292_out -> f277_out :|: TRUE 7.33/2.78 f277_in -> f292_in :|: TRUE 7.33/2.78 f296_out -> f291_out :|: TRUE 7.33/2.78 f295_out -> f296_in :|: TRUE 7.33/2.78 f291_in -> f295_in :|: TRUE 7.33/2.78 f336_out -> f337_in :|: TRUE 7.33/2.78 f296_in -> f336_in :|: TRUE 7.33/2.78 f337_out -> f296_out :|: TRUE 7.33/2.78 Start term: f211_in(T2) 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (92) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 7.33/2.78 Constructed simple dependency graph. 7.33/2.78 7.33/2.78 Simplified to the following IRSwTs: 7.33/2.78 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (93) 7.33/2.78 TRUE 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (94) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f339_in -> f341_in :|: TRUE 7.33/2.78 f341_out -> f339_out :|: TRUE 7.33/2.78 f342_out -> f339_out :|: TRUE 7.33/2.78 f339_in -> f342_in :|: TRUE 7.33/2.78 f296_out -> f291_out :|: TRUE 7.33/2.78 f295_out -> f296_in :|: TRUE 7.33/2.78 f291_in -> f295_in :|: TRUE 7.33/2.78 f340_out -> f338_out :|: TRUE 7.33/2.78 f338_in -> f340_in :|: TRUE 7.33/2.78 f339_out -> f338_out :|: TRUE 7.33/2.78 f338_in -> f339_in :|: TRUE 7.33/2.78 f337_out -> f344_out :|: TRUE 7.33/2.78 f344_in -> f337_in :|: TRUE 7.33/2.78 f267_out -> f263_out :|: TRUE 7.33/2.78 f263_in -> f267_in :|: TRUE 7.33/2.78 f263_out -> f295_out :|: TRUE 7.33/2.78 f295_in -> f263_in :|: TRUE 7.33/2.78 f338_out -> f337_out :|: TRUE 7.33/2.78 f337_in -> f338_in :|: TRUE 7.33/2.78 f277_in -> f291_in :|: TRUE 7.33/2.78 f291_out -> f277_out :|: TRUE 7.33/2.78 f292_out -> f277_out :|: TRUE 7.33/2.78 f277_in -> f292_in :|: TRUE 7.33/2.78 f267_in -> f268_in :|: TRUE 7.33/2.78 f268_out -> f267_out :|: TRUE 7.33/2.78 f267_in -> f269_in :|: TRUE 7.33/2.78 f269_out -> f267_out :|: TRUE 7.33/2.78 f336_in -> f263_in :|: TRUE 7.33/2.78 f263_out -> f336_out :|: TRUE 7.33/2.78 f345_out -> f340_out :|: TRUE 7.33/2.78 f340_in -> f345_in :|: TRUE 7.33/2.78 f344_out -> f340_out :|: TRUE 7.33/2.78 f340_in -> f344_in :|: TRUE 7.33/2.78 f277_out -> f269_out :|: TRUE 7.33/2.78 f269_in -> f275_in :|: TRUE 7.33/2.78 f275_out -> f269_out :|: TRUE 7.33/2.78 f269_in -> f277_in :|: TRUE 7.33/2.78 f336_out -> f337_in :|: TRUE 7.33/2.78 f296_in -> f336_in :|: TRUE 7.33/2.78 f337_out -> f296_out :|: TRUE 7.33/2.78 f341_in -> f341_out :|: TRUE 7.33/2.78 f211_in(T2) -> f215_in(T2) :|: TRUE 7.33/2.78 f215_out(x) -> f211_out(x) :|: TRUE 7.33/2.78 f216_out(x1) -> f215_out(x1) :|: TRUE 7.33/2.78 f215_in(x2) -> f216_in(x2) :|: TRUE 7.33/2.78 f215_in(x3) -> f217_in(x3) :|: TRUE 7.33/2.78 f217_out(x4) -> f215_out(x4) :|: TRUE 7.33/2.78 f223_out(x5) -> f217_out(x5) :|: TRUE 7.33/2.78 f217_in(x6) -> f223_in(x6) :|: TRUE 7.33/2.78 f225_out(x7) -> f217_out(x7) :|: TRUE 7.33/2.78 f217_in(x8) -> f225_in(x8) :|: TRUE 7.33/2.78 f225_in(T19) -> f249_in(T19) :|: TRUE 7.33/2.78 f249_out(x9) -> f225_out(x9) :|: TRUE 7.33/2.78 f225_in(x10) -> f251_in :|: TRUE 7.33/2.78 f251_out -> f225_out(x11) :|: TRUE 7.33/2.78 f263_out -> f264_in(x12) :|: TRUE 7.33/2.78 f264_out(x13) -> f249_out(x13) :|: TRUE 7.33/2.78 f249_in(x14) -> f263_in :|: TRUE 7.33/2.78 f264_in(x15) -> f398_in :|: TRUE 7.33/2.78 f398_out -> f399_in(x16) :|: TRUE 7.33/2.78 f399_out(x17) -> f264_out(x17) :|: TRUE 7.33/2.78 f398_in -> f263_in :|: TRUE 7.33/2.78 f263_out -> f398_out :|: TRUE 7.33/2.78 Start term: f211_in(T2) 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (95) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 7.33/2.78 Constructed simple dependency graph. 7.33/2.78 7.33/2.78 Simplified to the following IRSwTs: 7.33/2.78 7.33/2.78 intTRSProblem: 7.33/2.78 f291_in -> f295_in :|: TRUE 7.33/2.78 f263_in -> f267_in :|: TRUE 7.33/2.78 f295_in -> f263_in :|: TRUE 7.33/2.78 f277_in -> f291_in :|: TRUE 7.33/2.78 f267_in -> f269_in :|: TRUE 7.33/2.78 f269_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (96) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f291_in -> f295_in :|: TRUE 7.33/2.78 f263_in -> f267_in :|: TRUE 7.33/2.78 f295_in -> f263_in :|: TRUE 7.33/2.78 f277_in -> f291_in :|: TRUE 7.33/2.78 f267_in -> f269_in :|: TRUE 7.33/2.78 f269_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (97) IntTRSCompressionProof (EQUIVALENT) 7.33/2.78 Compressed rules. 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (98) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f277_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (99) IRSFormatTransformerProof (EQUIVALENT) 7.33/2.78 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (100) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f277_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (101) IRSwTTerminationDigraphProof (EQUIVALENT) 7.33/2.78 Constructed termination digraph! 7.33/2.78 Nodes: 7.33/2.78 (1) f277_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 Arcs: 7.33/2.78 (1) -> (1) 7.33/2.78 7.33/2.78 This digraph is fully evaluated! 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (102) 7.33/2.78 Obligation: 7.33/2.78 7.33/2.78 Termination digraph: 7.33/2.78 Nodes: 7.33/2.78 (1) f277_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 Arcs: 7.33/2.78 (1) -> (1) 7.33/2.78 7.33/2.78 This digraph is fully evaluated! 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (103) FilterProof (EQUIVALENT) 7.33/2.78 Used the following sort dictionary for filtering: 7.33/2.78 f277_in() 7.33/2.78 Replaced non-predefined constructor symbols by 0. 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (104) 7.33/2.78 Obligation: 7.33/2.78 Rules: 7.33/2.78 f277_in -> f277_in :|: TRUE 7.33/2.78 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (105) IntTRSPeriodicNontermProof (COMPLETE) 7.33/2.78 Normalized system to the following form: 7.33/2.78 f(pc) -> f(1) :|: pc = 1 && TRUE 7.33/2.78 Witness term starting non-terminating reduction: f(1) 7.33/2.78 ---------------------------------------- 7.33/2.78 7.33/2.78 (106) 7.33/2.78 NO 7.53/2.81 EOF