/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern color_map(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 2 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) TransformationProof [SOUND, 0 ms] (20) QDP (21) TransformationProof [EQUIVALENT, 0 ms] (22) QDP (23) NonTerminationLoopProof [COMPLETE, 0 ms] (24) NO (25) PiDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) PiDP (28) PiDPToQDPProof [SOUND, 0 ms] (29) QDP (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] (31) YES (32) PiDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) PiDP (35) PiDPToQDPProof [SOUND, 0 ms] (36) QDP (37) PrologToPiTRSProof [SOUND, 0 ms] (38) PiTRS (39) DependencyPairsProof [EQUIVALENT, 0 ms] (40) PiDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) AND (43) PiDP (44) UsableRulesProof [EQUIVALENT, 0 ms] (45) PiDP (46) PiDPToQDPProof [SOUND, 16 ms] (47) QDP (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] (49) YES (50) PiDP (51) UsableRulesProof [EQUIVALENT, 0 ms] (52) PiDP (53) PiDPToQDPProof [SOUND, 0 ms] (54) QDP (55) TransformationProof [SOUND, 0 ms] (56) QDP (57) TransformationProof [EQUIVALENT, 0 ms] (58) QDP (59) PiDP (60) UsableRulesProof [EQUIVALENT, 0 ms] (61) PiDP (62) PiDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) PiDP (65) PrologToDTProblemTransformerProof [SOUND, 43 ms] (66) TRIPLES (67) TriplesToPiDPProof [SOUND, 0 ms] (68) PiDP (69) DependencyGraphProof [EQUIVALENT, 0 ms] (70) AND (71) PiDP (72) UsableRulesProof [EQUIVALENT, 0 ms] (73) PiDP (74) PiDPToQDPProof [SOUND, 0 ms] (75) QDP (76) QDPSizeChangeProof [EQUIVALENT, 0 ms] (77) YES (78) PiDP (79) UsableRulesProof [EQUIVALENT, 0 ms] (80) PiDP (81) PiDPToQDPProof [SOUND, 0 ms] (82) QDP (83) TransformationProof [SOUND, 0 ms] (84) QDP (85) TransformationProof [EQUIVALENT, 0 ms] (86) QDP (87) PiDP (88) UsableRulesProof [EQUIVALENT, 0 ms] (89) PiDP (90) PiDP (91) PrologToTRSTransformerProof [SOUND, 30 ms] (92) QTRS (93) DependencyPairsProof [EQUIVALENT, 0 ms] (94) QDP (95) DependencyGraphProof [EQUIVALENT, 0 ms] (96) AND (97) QDP (98) UsableRulesProof [EQUIVALENT, 0 ms] (99) QDP (100) QDPSizeChangeProof [EQUIVALENT, 0 ms] (101) YES (102) QDP (103) NonTerminationLoopProof [COMPLETE, 1 ms] (104) NO (105) QDP (106) UsableRulesProof [EQUIVALENT, 0 ms] (107) QDP (108) QDPSizeChangeProof [EQUIVALENT, 0 ms] (109) YES (110) QDP (111) NonTerminationLoopProof [COMPLETE, 13 ms] (112) NO (113) PrologToIRSwTTransformerProof [SOUND, 30 ms] (114) AND (115) IRSwT (116) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (117) TRUE (118) IRSwT (119) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (120) TRUE (121) IRSwT (122) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (123) IRSwT (124) IntTRSCompressionProof [EQUIVALENT, 19 ms] (125) IRSwT (126) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (127) IRSwT (128) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (129) IRSwT (130) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 3 ms] (131) IRSwT (132) TempFilterProof [SOUND, 3 ms] (133) IRSwT (134) IRSwTToQDPProof [SOUND, 0 ms] (135) QDP (136) QDPSizeChangeProof [EQUIVALENT, 0 ms] (137) YES (138) IRSwT (139) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (140) IRSwT (141) IntTRSCompressionProof [EQUIVALENT, 25 ms] (142) IRSwT (143) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (144) IRSwT (145) IRSwTTerminationDigraphProof [EQUIVALENT, 79 ms] (146) IRSwT (147) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (148) IRSwT ---------------------------------------- (0) Obligation: Clauses: color_map(.(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)). color_map([], Colors). color_region(region(Name, Color, Neighbors), Colors) :- ','(select(Color, Colors, Colors1), members(Neighbors, Colors1)). select(X, .(X, Xs), Xs). select(X, .(Y, Ys), .(Y, Zs)) :- select(X, Ys, Zs). members(.(X, Xs), Ys) :- ','(member(X, Ys), members(Xs, Ys)). members([], Ys). member(X, .(X, X1)). member(X, .(X2, Xs)) :- member(X, Xs). Query: color_map(a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: color_map_in_2: (f,b) color_region_in_2: (f,b) select_in_3: (f,b,f) members_in_2: (f,b) member_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> COLOR_REGION_IN_AG(Region, Colors) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> U3_AG(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> SELECT_IN_AGA(Color, Colors, Colors1) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> U5_AGA(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_AG(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> MEMBERS_IN_AG(Neighbors, Colors1) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) MEMBERS_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(X2, Xs)) -> U8_AG(X, X2, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_AG(X, Xs, Ys, members_in_ag(Xs, Ys)) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_AG(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) COLOR_REGION_IN_AG(x1, x2) = COLOR_REGION_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x5) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x2, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x2, x5) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U8_AG(x1, x2, x3, x4) = U8_AG(x4) U7_AG(x1, x2, x3, x4) = U7_AG(x1, x4) U2_AG(x1, x2, x3, x4) = U2_AG(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> COLOR_REGION_IN_AG(Region, Colors) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> U3_AG(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> SELECT_IN_AGA(Color, Colors, Colors1) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> U5_AGA(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_AG(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> MEMBERS_IN_AG(Neighbors, Colors1) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) MEMBERS_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(X2, Xs)) -> U8_AG(X, X2, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_AG(X, Xs, Ys, members_in_ag(Xs, Ys)) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_AG(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) COLOR_REGION_IN_AG(x1, x2) = COLOR_REGION_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x5) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x2, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x2, x5) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U8_AG(x1, x2, x3, x4) = U8_AG(x4) U7_AG(x1, x2, x3, x4) = U7_AG(x1, x4) U2_AG(x1, x2, x3, x4) = U2_AG(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(X2, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *MEMBER_IN_AG(.(X2, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AG(Ys, member_out_ag(X)) -> MEMBERS_IN_AG(Ys) MEMBERS_IN_AG(Ys) -> U6_AG(Ys, member_in_ag(Ys)) The TRS R consists of the following rules: member_in_ag(.(X, X1)) -> member_out_ag(X) member_in_ag(.(X2, Xs)) -> U8_ag(member_in_ag(Xs)) U8_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U8_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) TransformationProof (SOUND) By narrowing [LPAR04] the rule MEMBERS_IN_AG(Ys) -> U6_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]: (MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0)),MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0))) (MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(member_in_ag(x1))),MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(member_in_ag(x1)))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AG(Ys, member_out_ag(X)) -> MEMBERS_IN_AG(Ys) MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0)) MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(member_in_ag(x1))) The TRS R consists of the following rules: member_in_ag(.(X, X1)) -> member_out_ag(X) member_in_ag(.(X2, Xs)) -> U8_ag(member_in_ag(Xs)) U8_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U8_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U6_AG(Ys, member_out_ag(X)) -> MEMBERS_IN_AG(Ys) we obtained the following new rules [LPAR04]: (U6_AG(.(z0, z1), member_out_ag(z0)) -> MEMBERS_IN_AG(.(z0, z1)),U6_AG(.(z0, z1), member_out_ag(z0)) -> MEMBERS_IN_AG(.(z0, z1))) (U6_AG(.(z0, z1), member_out_ag(x1)) -> MEMBERS_IN_AG(.(z0, z1)),U6_AG(.(z0, z1), member_out_ag(x1)) -> MEMBERS_IN_AG(.(z0, z1))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0)) MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(member_in_ag(x1))) U6_AG(.(z0, z1), member_out_ag(z0)) -> MEMBERS_IN_AG(.(z0, z1)) U6_AG(.(z0, z1), member_out_ag(x1)) -> MEMBERS_IN_AG(.(z0, z1)) The TRS R consists of the following rules: member_in_ag(.(X, X1)) -> member_out_ag(X) member_in_ag(.(X2, Xs)) -> U8_ag(member_in_ag(Xs)) U8_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U8_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = U6_AG(.(z0, z1), member_out_ag(z0)) evaluates to t =U6_AG(.(z0, z1), member_out_ag(z0)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U6_AG(.(z0, z1), member_out_ag(z0)) -> MEMBERS_IN_AG(.(z0, z1)) with rule U6_AG(.(z0', z1'), member_out_ag(z0')) -> MEMBERS_IN_AG(.(z0', z1')) at position [] and matcher [z0' / z0, z1' / z1] MEMBERS_IN_AG(.(z0, z1)) -> U6_AG(.(z0, z1), member_out_ag(z0)) with rule MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0)) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (24) NO ---------------------------------------- (25) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (27) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: SELECT_IN_AGA(.(Y, Ys)) -> SELECT_IN_AGA(Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *SELECT_IN_AGA(.(Y, Ys)) -> SELECT_IN_AGA(Ys) The graph contains the following edges 1 > 1 ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) The TRS R consists of the following rules: color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U8_ag(x1, x2, x3, x4) = U8_ag(x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x4) members_out_ag(x1, x2) = members_out_ag(x1) color_region_out_ag(x1, x2) = color_region_out_ag(x1) region(x1, x2, x3) = region(x2, x3) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AG(Colors, color_region_out_ag(Region)) -> COLOR_MAP_IN_AG(Colors) COLOR_MAP_IN_AG(Colors) -> U1_AG(Colors, color_region_in_ag(Colors)) The TRS R consists of the following rules: color_region_in_ag(Colors) -> U3_ag(select_in_aga(Colors)) U3_ag(select_out_aga(Color, Colors1)) -> U4_ag(Color, members_in_ag(Colors1)) select_in_aga(.(X, Xs)) -> select_out_aga(X, Xs) select_in_aga(.(Y, Ys)) -> U5_aga(Y, select_in_aga(Ys)) U4_ag(Color, members_out_ag(Neighbors)) -> color_region_out_ag(region(Color, Neighbors)) U5_aga(Y, select_out_aga(X, Zs)) -> select_out_aga(X, .(Y, Zs)) members_in_ag(Ys) -> U6_ag(Ys, member_in_ag(Ys)) members_in_ag(Ys) -> members_out_ag([]) U6_ag(Ys, member_out_ag(X)) -> U7_ag(X, members_in_ag(Ys)) member_in_ag(.(X, X1)) -> member_out_ag(X) member_in_ag(.(X2, Xs)) -> U8_ag(member_in_ag(Xs)) U7_ag(X, members_out_ag(Xs)) -> members_out_ag(.(X, Xs)) U8_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: color_region_in_ag(x0) U3_ag(x0) select_in_aga(x0) U4_ag(x0, x1) U5_aga(x0, x1) members_in_ag(x0) U6_ag(x0, x1) member_in_ag(x0) U7_ag(x0, x1) U8_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: color_map_in_2: (f,b) color_region_in_2: (f,b) select_in_3: (f,b,f) members_in_2: (f,b) member_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (38) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) ---------------------------------------- (39) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> COLOR_REGION_IN_AG(Region, Colors) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> U3_AG(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> SELECT_IN_AGA(Color, Colors, Colors1) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> U5_AGA(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_AG(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> MEMBERS_IN_AG(Neighbors, Colors1) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) MEMBERS_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(X2, Xs)) -> U8_AG(X, X2, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_AG(X, Xs, Ys, members_in_ag(Xs, Ys)) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_AG(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) COLOR_REGION_IN_AG(x1, x2) = COLOR_REGION_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x2, x3, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x2, x4, x5) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U8_AG(x1, x2, x3, x4) = U8_AG(x2, x3, x4) U7_AG(x1, x2, x3, x4) = U7_AG(x1, x3, x4) U2_AG(x1, x2, x3, x4) = U2_AG(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) Obligation: Pi DP problem: The TRS P consists of the following rules: COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> COLOR_REGION_IN_AG(Region, Colors) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> U3_AG(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) COLOR_REGION_IN_AG(region(Name, Color, Neighbors), Colors) -> SELECT_IN_AGA(Color, Colors, Colors1) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> U5_AGA(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_AG(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) U3_AG(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> MEMBERS_IN_AG(Neighbors, Colors1) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) MEMBERS_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(X2, Xs)) -> U8_AG(X, X2, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_AG(X, Xs, Ys, members_in_ag(Xs, Ys)) U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_AG(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) COLOR_REGION_IN_AG(x1, x2) = COLOR_REGION_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x2, x3, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x2, x4, x5) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U8_AG(x1, x2, x3, x4) = U8_AG(x2, x3, x4) U7_AG(x1, x2, x3, x4) = U7_AG(x1, x3, x4) U2_AG(x1, x2, x3, x4) = U2_AG(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes. ---------------------------------------- (42) Complex Obligation (AND) ---------------------------------------- (43) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (44) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (45) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X2, Xs)) -> MEMBER_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (46) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(X2, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (48) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *MEMBER_IN_AG(.(X2, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (49) YES ---------------------------------------- (50) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (51) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Xs, Ys) MEMBERS_IN_AG(.(X, Xs), Ys) -> U6_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) MEMBERS_IN_AG(x1, x2) = MEMBERS_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AG(Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Ys) MEMBERS_IN_AG(Ys) -> U6_AG(Ys, member_in_ag(Ys)) The TRS R consists of the following rules: member_in_ag(.(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(.(X2, Xs)) -> U8_ag(X2, Xs, member_in_ag(Xs)) U8_ag(X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The set Q consists of the following terms: member_in_ag(x0) U8_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (SOUND) By narrowing [LPAR04] the rule MEMBERS_IN_AG(Ys) -> U6_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]: (MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))),MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0, .(x0, x1)))) (MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(x0, x1, member_in_ag(x1))),MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(x0, x1, member_in_ag(x1)))) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AG(Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Ys) MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))) MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(x0, x1, member_in_ag(x1))) The TRS R consists of the following rules: member_in_ag(.(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(.(X2, Xs)) -> U8_ag(X2, Xs, member_in_ag(Xs)) U8_ag(X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The set Q consists of the following terms: member_in_ag(x0) U8_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (57) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U6_AG(Ys, member_out_ag(X, Ys)) -> MEMBERS_IN_AG(Ys) we obtained the following new rules [LPAR04]: (U6_AG(.(z0, z1), member_out_ag(z0, .(z0, z1))) -> MEMBERS_IN_AG(.(z0, z1)),U6_AG(.(z0, z1), member_out_ag(z0, .(z0, z1))) -> MEMBERS_IN_AG(.(z0, z1))) (U6_AG(.(z0, z1), member_out_ag(x1, .(z0, z1))) -> MEMBERS_IN_AG(.(z0, z1)),U6_AG(.(z0, z1), member_out_ag(x1, .(z0, z1))) -> MEMBERS_IN_AG(.(z0, z1))) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))) MEMBERS_IN_AG(.(x0, x1)) -> U6_AG(.(x0, x1), U8_ag(x0, x1, member_in_ag(x1))) U6_AG(.(z0, z1), member_out_ag(z0, .(z0, z1))) -> MEMBERS_IN_AG(.(z0, z1)) U6_AG(.(z0, z1), member_out_ag(x1, .(z0, z1))) -> MEMBERS_IN_AG(.(z0, z1)) The TRS R consists of the following rules: member_in_ag(.(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(.(X2, Xs)) -> U8_ag(X2, Xs, member_in_ag(Xs)) U8_ag(X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The set Q consists of the following terms: member_in_ag(x0) U8_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (59) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (60) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (61) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECT_IN_AGA(X, .(Y, Ys), .(Y, Zs)) -> SELECT_IN_AGA(X, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SELECT_IN_AGA(x1, x2, x3) = SELECT_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) The TRS R consists of the following rules: color_map_in_ag(.(Region, Regions), Colors) -> U1_ag(Region, Regions, Colors, color_region_in_ag(Region, Colors)) color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U1_ag(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> U2_ag(Region, Regions, Colors, color_map_in_ag(Regions, Colors)) color_map_in_ag([], Colors) -> color_map_out_ag([], Colors) U2_ag(Region, Regions, Colors, color_map_out_ag(Regions, Colors)) -> color_map_out_ag(.(Region, Regions), Colors) The argument filtering Pi contains the following mapping: color_map_in_ag(x1, x2) = color_map_in_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) U2_ag(x1, x2, x3, x4) = U2_ag(x1, x3, x4) color_map_out_ag(x1, x2) = color_map_out_ag(x1, x2) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (64) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(Region, Regions, Colors, color_region_out_ag(Region, Colors)) -> COLOR_MAP_IN_AG(Regions, Colors) COLOR_MAP_IN_AG(.(Region, Regions), Colors) -> U1_AG(Region, Regions, Colors, color_region_in_ag(Region, Colors)) The TRS R consists of the following rules: color_region_in_ag(region(Name, Color, Neighbors), Colors) -> U3_ag(Name, Color, Neighbors, Colors, select_in_aga(Color, Colors, Colors1)) U3_ag(Name, Color, Neighbors, Colors, select_out_aga(Color, Colors, Colors1)) -> U4_ag(Name, Color, Neighbors, Colors, members_in_ag(Neighbors, Colors1)) select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs) select_in_aga(X, .(Y, Ys), .(Y, Zs)) -> U5_aga(X, Y, Ys, Zs, select_in_aga(X, Ys, Zs)) U4_ag(Name, Color, Neighbors, Colors, members_out_ag(Neighbors, Colors1)) -> color_region_out_ag(region(Name, Color, Neighbors), Colors) U5_aga(X, Y, Ys, Zs, select_out_aga(X, Ys, Zs)) -> select_out_aga(X, .(Y, Ys), .(Y, Zs)) members_in_ag(.(X, Xs), Ys) -> U6_ag(X, Xs, Ys, member_in_ag(X, Ys)) members_in_ag([], Ys) -> members_out_ag([], Ys) U6_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U7_ag(X, Xs, Ys, members_in_ag(Xs, Ys)) member_in_ag(X, .(X, X1)) -> member_out_ag(X, .(X, X1)) member_in_ag(X, .(X2, Xs)) -> U8_ag(X, X2, Xs, member_in_ag(X, Xs)) U7_ag(X, Xs, Ys, members_out_ag(Xs, Ys)) -> members_out_ag(.(X, Xs), Ys) U8_ag(X, X2, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X2, Xs)) The argument filtering Pi contains the following mapping: color_region_in_ag(x1, x2) = color_region_in_ag(x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) select_in_aga(x1, x2, x3) = select_in_aga(x2) .(x1, x2) = .(x1, x2) select_out_aga(x1, x2, x3) = select_out_aga(x1, x2, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x3, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x2, x4, x5) members_in_ag(x1, x2) = members_in_ag(x2) U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U8_ag(x1, x2, x3, x4) = U8_ag(x2, x3, x4) U7_ag(x1, x2, x3, x4) = U7_ag(x1, x3, x4) members_out_ag(x1, x2) = members_out_ag(x1, x2) color_region_out_ag(x1, x2) = color_region_out_ag(x1, x2) region(x1, x2, x3) = region(x2, x3) COLOR_MAP_IN_AG(x1, x2) = COLOR_MAP_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (65) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(color_map (. Region Regions) Colors)", "(',' (color_region Region Colors) (color_map Regions Colors))" ], [ "(color_map ([]) Colors)", null ], [ "(color_region (region Name Color Neighbors) Colors)", "(',' (select Color Colors Colors1) (members Neighbors Colors1))" ], [ "(select X (. X Xs) Xs)", null ], [ "(select X (. Y Ys) (. Y Zs))", "(select X Ys Zs)" ], [ "(members (. X Xs) Ys)", "(',' (member X Ys) (members Xs Ys))" ], [ "(members ([]) Ys)", null ], [ "(member X (. X X1))", null ], [ "(member X (. X2 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "25": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (color_region T9 T8) (color_map T10 T8))" }, { "clause": 1, "scope": 1, "term": "(color_map T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": 1, "scope": 1, "term": "(color_map T1 T2)" }], "kb": { "nonunifying": [[ "(color_map T1 T2)", "(color_map (. X6 X7) X8)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [ "X6", "X7", "X8" ], "exprvars": [] } }, "27": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (color_region T9 T8) (color_map T10 T8))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(color_map T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "type": "Nodes", "470": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "350": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T64 T63 X64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T63"], "free": ["X64"], "exprvars": [] } }, "471": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "351": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "450": { "goal": [{ "clause": 6, "scope": 4, "term": "(members T39 T38)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T38"], "free": [], "exprvars": [] } }, "472": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "451": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T91 T90) (members T92 T90))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [], "exprvars": [] } }, "473": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "452": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "453": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T91 T90)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [], "exprvars": [] } }, "454": { "goal": [{ "clause": -1, "scope": -1, "term": "(members T97 T90)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [], "exprvars": [] } }, "334": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T31 T30 X31)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": ["X31"], "exprvars": [] } }, "455": { "goal": [ { "clause": 7, "scope": 5, "term": "(member T91 T90)" }, { "clause": 8, "scope": 5, "term": "(member T91 T90)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [], "exprvars": [] } }, "335": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (members T39 T38) (color_map T40 T30))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T38" ], "free": [], "exprvars": [] } }, "456": { "goal": [{ "clause": 7, "scope": 5, "term": "(member T91 T90)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [], "exprvars": [] } }, "458": { "goal": [{ "clause": 8, "scope": 5, "term": "(member T91 T90)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [], "exprvars": [] } }, "338": { "goal": [ { "clause": 3, "scope": 3, "term": "(select T31 T30 X31)" }, { "clause": 4, "scope": 3, "term": "(select T31 T30 X31)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": ["X31"], "exprvars": [] } }, "459": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "339": { "goal": [{ "clause": 3, "scope": 3, "term": "(select T31 T30 X31)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": ["X31"], "exprvars": [] } }, "32": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (color_region T9 T8) (color_map T10 T8))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "33": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(color_map T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "38": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (select T31 T30 X31) (members T32 X31)) (color_map T33 T30))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": ["X31"], "exprvars": [] } }, "39": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "460": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "340": { "goal": [{ "clause": 4, "scope": 3, "term": "(select T31 T30 X31)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": ["X31"], "exprvars": [] } }, "461": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "462": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T121 T120)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T120"], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_map T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "463": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [ { "clause": 0, "scope": 1, "term": "(color_map T1 T2)" }, { "clause": 1, "scope": 1, "term": "(color_map T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "464": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "344": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "465": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "345": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "444": { "goal": [{ "clause": -1, "scope": -1, "term": "(members T39 T38)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T38"], "free": [], "exprvars": [] } }, "466": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "346": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "445": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_map T72 T30)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": [], "exprvars": [] } }, "467": { "goal": [{ "clause": 1, "scope": 1, "term": "(color_map T1 T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "468": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "469": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "448": { "goal": [ { "clause": 5, "scope": 4, "term": "(members T39 T38)" }, { "clause": 6, "scope": 4, "term": "(members T39 T38)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T38"], "free": [], "exprvars": [] } }, "449": { "goal": [{ "clause": 5, "scope": 4, "term": "(members T39 T38)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T38"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 25, "label": "EVAL with clause\ncolor_map(.(X6, X7), X8) :- ','(color_region(X6, X8), color_map(X7, X8)).\nand substitutionX6 -> T9,\nX7 -> T10,\nT1 -> .(T9, T10),\nT2 -> T8,\nX8 -> T8,\nT6 -> T9,\nT7 -> T10" }, { "from": 2, "to": 26, "label": "EVAL-BACKTRACK" }, { "from": 25, "to": 27, "label": "CASE" }, { "from": 26, "to": 471, "label": "EVAL with clause\ncolor_map([], X134).\nand substitutionT1 -> [],\nT2 -> T139,\nX134 -> T139" }, { "from": 26, "to": 472, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 32, "label": "PARALLEL" }, { "from": 27, "to": 33, "label": "PARALLEL" }, { "from": 32, "to": 38, "label": "EVAL with clause\ncolor_region(region(X27, X28, X29), X30) :- ','(select(X28, X30, X31), members(X29, X31)).\nand substitutionX27 -> T27,\nX28 -> T31,\nX29 -> T32,\nT9 -> region(T27, T31, T32),\nT8 -> T30,\nX30 -> T30,\nT28 -> T31,\nT29 -> T32,\nT10 -> T33" }, { "from": 32, "to": 39, "label": "EVAL-BACKTRACK" }, { "from": 33, "to": 467, "label": "FAILURE" }, { "from": 38, "to": 334, "label": "SPLIT 1" }, { "from": 38, "to": 335, "label": "SPLIT 2\nnew knowledge:\nT31 is ground\nT30 is ground\nT38 is ground\nreplacements:X31 -> T38,\nT32 -> T39,\nT33 -> T40" }, { "from": 334, "to": 338, "label": "CASE" }, { "from": 335, "to": 444, "label": "SPLIT 1" }, { "from": 335, "to": 445, "label": "SPLIT 2\nnew knowledge:\nT39 is ground\nT38 is ground\nreplacements:T40 -> T72" }, { "from": 338, "to": 339, "label": "PARALLEL" }, { "from": 338, "to": 340, "label": "PARALLEL" }, { "from": 339, "to": 344, "label": "EVAL with clause\nselect(X48, .(X48, X49), X49).\nand substitutionT31 -> T53,\nX48 -> T53,\nX49 -> T54,\nT30 -> .(T53, T54),\nX31 -> T54" }, { "from": 339, "to": 345, "label": "EVAL-BACKTRACK" }, { "from": 340, "to": 350, "label": "EVAL with clause\nselect(X60, .(X61, X62), .(X61, X63)) :- select(X60, X62, X63).\nand substitutionT31 -> T64,\nX60 -> T64,\nX61 -> T62,\nX62 -> T63,\nT30 -> .(T62, T63),\nX63 -> X64,\nX31 -> .(T62, X64),\nT61 -> T64" }, { "from": 340, "to": 351, "label": "EVAL-BACKTRACK" }, { "from": 344, "to": 346, "label": "SUCCESS" }, { "from": 350, "to": 334, "label": "INSTANCE with matching:\nT31 -> T64\nT30 -> T63\nX31 -> X64" }, { "from": 444, "to": 448, "label": "CASE" }, { "from": 445, "to": 1, "label": "INSTANCE with matching:\nT1 -> T72\nT2 -> T30" }, { "from": 448, "to": 449, "label": "PARALLEL" }, { "from": 448, "to": 450, "label": "PARALLEL" }, { "from": 449, "to": 451, "label": "EVAL with clause\nmembers(.(X87, X88), X89) :- ','(member(X87, X89), members(X88, X89)).\nand substitutionX87 -> T91,\nX88 -> T92,\nT39 -> .(T91, T92),\nT38 -> T90,\nX89 -> T90,\nT88 -> T91,\nT89 -> T92" }, { "from": 449, "to": 452, "label": "EVAL-BACKTRACK" }, { "from": 450, "to": 464, "label": "EVAL with clause\nmembers([], X126).\nand substitutionT39 -> [],\nT38 -> T131,\nX126 -> T131" }, { "from": 450, "to": 465, "label": "EVAL-BACKTRACK" }, { "from": 451, "to": 453, "label": "SPLIT 1" }, { "from": 451, "to": 454, "label": "SPLIT 2\nnew knowledge:\nT91 is ground\nT90 is ground\nreplacements:T92 -> T97" }, { "from": 453, "to": 455, "label": "CASE" }, { "from": 454, "to": 444, "label": "INSTANCE with matching:\nT39 -> T97\nT38 -> T90" }, { "from": 455, "to": 456, "label": "PARALLEL" }, { "from": 455, "to": 458, "label": "PARALLEL" }, { "from": 456, "to": 459, "label": "EVAL with clause\nmember(X106, .(X106, X107)).\nand substitutionT91 -> T110,\nX106 -> T110,\nX107 -> T111,\nT90 -> .(T110, T111)" }, { "from": 456, "to": 460, "label": "EVAL-BACKTRACK" }, { "from": 458, "to": 462, "label": "EVAL with clause\nmember(X114, .(X115, X116)) :- member(X114, X116).\nand substitutionT91 -> T121,\nX114 -> T121,\nX115 -> T119,\nX116 -> T120,\nT90 -> .(T119, T120),\nT118 -> T121" }, { "from": 458, "to": 463, "label": "EVAL-BACKTRACK" }, { "from": 459, "to": 461, "label": "SUCCESS" }, { "from": 462, "to": 453, "label": "INSTANCE with matching:\nT91 -> T121\nT90 -> T120" }, { "from": 464, "to": 466, "label": "SUCCESS" }, { "from": 467, "to": 468, "label": "EVAL with clause\ncolor_map([], X132).\nand substitutionT1 -> [],\nT8 -> T137,\nX132 -> T137" }, { "from": 467, "to": 469, "label": "EVAL-BACKTRACK" }, { "from": 468, "to": 470, "label": "SUCCESS" }, { "from": 471, "to": 473, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (66) Obligation: Triples: selectA(X1, .(X2, X3), .(X2, X4)) :- selectA(X1, X3, X4). membersC(.(X1, X2), X3) :- memberD(X1, X3). membersC(.(X1, X2), X3) :- ','(membercD(X1, X3), membersC(X2, X3)). memberD(X1, .(X2, X3)) :- memberD(X1, X3). color_mapB(.(region(X1, X2, X3), X4), X5) :- selectA(X2, X5, X6). color_mapB(.(region(X1, X2, X3), X4), X5) :- ','(selectcA(X2, X5, X6), membersC(X3, X6)). color_mapB(.(region(X1, X2, X3), X4), X5) :- ','(selectcA(X2, X5, X6), ','(memberscC(X3, X6), color_mapB(X4, X5))). Clauses: selectcA(X1, .(X1, X2), X2). selectcA(X1, .(X2, X3), .(X2, X4)) :- selectcA(X1, X3, X4). color_mapcB(.(region(X1, X2, X3), X4), X5) :- ','(selectcA(X2, X5, X6), ','(memberscC(X3, X6), color_mapcB(X4, X5))). color_mapcB([], X1). color_mapcB([], X1). memberscC(.(X1, X2), X3) :- ','(membercD(X1, X3), memberscC(X2, X3)). memberscC([], X1). membercD(X1, .(X1, X2)). membercD(X1, .(X2, X3)) :- membercD(X1, X3). Afs: color_mapB(x1, x2) = color_mapB(x2) ---------------------------------------- (67) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: color_mapB_in_2: (f,b) selectA_in_3: (f,b,f) selectcA_in_3: (f,b,f) membersC_in_2: (f,b) memberD_in_2: (f,b) membercD_in_2: (f,b) memberscC_in_2: (f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> U6_AG(X1, X2, X3, X4, X5, selectA_in_aga(X2, X5, X6)) COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> SELECTA_IN_AGA(X2, X5, X6) SELECTA_IN_AGA(X1, .(X2, X3), .(X2, X4)) -> U1_AGA(X1, X2, X3, X4, selectA_in_aga(X1, X3, X4)) SELECTA_IN_AGA(X1, .(X2, X3), .(X2, X4)) -> SELECTA_IN_AGA(X1, X3, X4) COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> U7_AG(X1, X2, X3, X4, X5, selectcA_in_aga(X2, X5, X6)) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> U8_AG(X1, X2, X3, X4, X5, membersC_in_ag(X3, X6)) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> MEMBERSC_IN_AG(X3, X6) MEMBERSC_IN_AG(.(X1, X2), X3) -> U2_AG(X1, X2, X3, memberD_in_ag(X1, X3)) MEMBERSC_IN_AG(.(X1, X2), X3) -> MEMBERD_IN_AG(X1, X3) MEMBERD_IN_AG(X1, .(X2, X3)) -> U5_AG(X1, X2, X3, memberD_in_ag(X1, X3)) MEMBERD_IN_AG(X1, .(X2, X3)) -> MEMBERD_IN_AG(X1, X3) MEMBERSC_IN_AG(.(X1, X2), X3) -> U3_AG(X1, X2, X3, membercD_in_ag(X1, X3)) U3_AG(X1, X2, X3, membercD_out_ag(X1, X3)) -> U4_AG(X1, X2, X3, membersC_in_ag(X2, X3)) U3_AG(X1, X2, X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X2, X3) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> U9_AG(X1, X2, X3, X4, X5, memberscC_in_ag(X3, X6)) U9_AG(X1, X2, X3, X4, X5, memberscC_out_ag(X3, X6)) -> U10_AG(X1, X2, X3, X4, X5, color_mapB_in_ag(X4, X5)) U9_AG(X1, X2, X3, X4, X5, memberscC_out_ag(X3, X6)) -> COLOR_MAPB_IN_AG(X4, X5) The TRS R consists of the following rules: selectcA_in_aga(X1, .(X1, X2), X2) -> selectcA_out_aga(X1, .(X1, X2), X2) selectcA_in_aga(X1, .(X2, X3), .(X2, X4)) -> U12_aga(X1, X2, X3, X4, selectcA_in_aga(X1, X3, X4)) U12_aga(X1, X2, X3, X4, selectcA_out_aga(X1, X3, X4)) -> selectcA_out_aga(X1, .(X2, X3), .(X2, X4)) membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) memberscC_in_ag(.(X1, X2), X3) -> U16_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U16_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> U17_ag(X1, X2, X3, memberscC_in_ag(X2, X3)) memberscC_in_ag([], X1) -> memberscC_out_ag([], X1) U17_ag(X1, X2, X3, memberscC_out_ag(X2, X3)) -> memberscC_out_ag(.(X1, X2), X3) The argument filtering Pi contains the following mapping: color_mapB_in_ag(x1, x2) = color_mapB_in_ag(x2) selectA_in_aga(x1, x2, x3) = selectA_in_aga(x2) .(x1, x2) = .(x1, x2) region(x1, x2, x3) = region(x2, x3) selectcA_in_aga(x1, x2, x3) = selectcA_in_aga(x2) selectcA_out_aga(x1, x2, x3) = selectcA_out_aga(x1, x2, x3) U12_aga(x1, x2, x3, x4, x5) = U12_aga(x2, x3, x5) membersC_in_ag(x1, x2) = membersC_in_ag(x2) memberD_in_ag(x1, x2) = memberD_in_ag(x2) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) memberscC_in_ag(x1, x2) = memberscC_in_ag(x2) U16_ag(x1, x2, x3, x4) = U16_ag(x3, x4) U17_ag(x1, x2, x3, x4) = U17_ag(x1, x3, x4) memberscC_out_ag(x1, x2) = memberscC_out_ag(x1, x2) COLOR_MAPB_IN_AG(x1, x2) = COLOR_MAPB_IN_AG(x2) U6_AG(x1, x2, x3, x4, x5, x6) = U6_AG(x5, x6) SELECTA_IN_AGA(x1, x2, x3) = SELECTA_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x2, x3, x5) U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x5, x6) U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x5, x6) MEMBERSC_IN_AG(x1, x2) = MEMBERSC_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) MEMBERD_IN_AG(x1, x2) = MEMBERD_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4) U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) U9_AG(x1, x2, x3, x4, x5, x6) = U9_AG(x5, x6) U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x5, x6) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (68) Obligation: Pi DP problem: The TRS P consists of the following rules: COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> U6_AG(X1, X2, X3, X4, X5, selectA_in_aga(X2, X5, X6)) COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> SELECTA_IN_AGA(X2, X5, X6) SELECTA_IN_AGA(X1, .(X2, X3), .(X2, X4)) -> U1_AGA(X1, X2, X3, X4, selectA_in_aga(X1, X3, X4)) SELECTA_IN_AGA(X1, .(X2, X3), .(X2, X4)) -> SELECTA_IN_AGA(X1, X3, X4) COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> U7_AG(X1, X2, X3, X4, X5, selectcA_in_aga(X2, X5, X6)) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> U8_AG(X1, X2, X3, X4, X5, membersC_in_ag(X3, X6)) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> MEMBERSC_IN_AG(X3, X6) MEMBERSC_IN_AG(.(X1, X2), X3) -> U2_AG(X1, X2, X3, memberD_in_ag(X1, X3)) MEMBERSC_IN_AG(.(X1, X2), X3) -> MEMBERD_IN_AG(X1, X3) MEMBERD_IN_AG(X1, .(X2, X3)) -> U5_AG(X1, X2, X3, memberD_in_ag(X1, X3)) MEMBERD_IN_AG(X1, .(X2, X3)) -> MEMBERD_IN_AG(X1, X3) MEMBERSC_IN_AG(.(X1, X2), X3) -> U3_AG(X1, X2, X3, membercD_in_ag(X1, X3)) U3_AG(X1, X2, X3, membercD_out_ag(X1, X3)) -> U4_AG(X1, X2, X3, membersC_in_ag(X2, X3)) U3_AG(X1, X2, X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X2, X3) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> U9_AG(X1, X2, X3, X4, X5, memberscC_in_ag(X3, X6)) U9_AG(X1, X2, X3, X4, X5, memberscC_out_ag(X3, X6)) -> U10_AG(X1, X2, X3, X4, X5, color_mapB_in_ag(X4, X5)) U9_AG(X1, X2, X3, X4, X5, memberscC_out_ag(X3, X6)) -> COLOR_MAPB_IN_AG(X4, X5) The TRS R consists of the following rules: selectcA_in_aga(X1, .(X1, X2), X2) -> selectcA_out_aga(X1, .(X1, X2), X2) selectcA_in_aga(X1, .(X2, X3), .(X2, X4)) -> U12_aga(X1, X2, X3, X4, selectcA_in_aga(X1, X3, X4)) U12_aga(X1, X2, X3, X4, selectcA_out_aga(X1, X3, X4)) -> selectcA_out_aga(X1, .(X2, X3), .(X2, X4)) membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) memberscC_in_ag(.(X1, X2), X3) -> U16_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U16_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> U17_ag(X1, X2, X3, memberscC_in_ag(X2, X3)) memberscC_in_ag([], X1) -> memberscC_out_ag([], X1) U17_ag(X1, X2, X3, memberscC_out_ag(X2, X3)) -> memberscC_out_ag(.(X1, X2), X3) The argument filtering Pi contains the following mapping: color_mapB_in_ag(x1, x2) = color_mapB_in_ag(x2) selectA_in_aga(x1, x2, x3) = selectA_in_aga(x2) .(x1, x2) = .(x1, x2) region(x1, x2, x3) = region(x2, x3) selectcA_in_aga(x1, x2, x3) = selectcA_in_aga(x2) selectcA_out_aga(x1, x2, x3) = selectcA_out_aga(x1, x2, x3) U12_aga(x1, x2, x3, x4, x5) = U12_aga(x2, x3, x5) membersC_in_ag(x1, x2) = membersC_in_ag(x2) memberD_in_ag(x1, x2) = memberD_in_ag(x2) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) memberscC_in_ag(x1, x2) = memberscC_in_ag(x2) U16_ag(x1, x2, x3, x4) = U16_ag(x3, x4) U17_ag(x1, x2, x3, x4) = U17_ag(x1, x3, x4) memberscC_out_ag(x1, x2) = memberscC_out_ag(x1, x2) COLOR_MAPB_IN_AG(x1, x2) = COLOR_MAPB_IN_AG(x2) U6_AG(x1, x2, x3, x4, x5, x6) = U6_AG(x5, x6) SELECTA_IN_AGA(x1, x2, x3) = SELECTA_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x2, x3, x5) U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x5, x6) U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x5, x6) MEMBERSC_IN_AG(x1, x2) = MEMBERSC_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) MEMBERD_IN_AG(x1, x2) = MEMBERD_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4) U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) U9_AG(x1, x2, x3, x4, x5, x6) = U9_AG(x5, x6) U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (69) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes. ---------------------------------------- (70) Complex Obligation (AND) ---------------------------------------- (71) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERD_IN_AG(X1, .(X2, X3)) -> MEMBERD_IN_AG(X1, X3) The TRS R consists of the following rules: selectcA_in_aga(X1, .(X1, X2), X2) -> selectcA_out_aga(X1, .(X1, X2), X2) selectcA_in_aga(X1, .(X2, X3), .(X2, X4)) -> U12_aga(X1, X2, X3, X4, selectcA_in_aga(X1, X3, X4)) U12_aga(X1, X2, X3, X4, selectcA_out_aga(X1, X3, X4)) -> selectcA_out_aga(X1, .(X2, X3), .(X2, X4)) membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) memberscC_in_ag(.(X1, X2), X3) -> U16_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U16_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> U17_ag(X1, X2, X3, memberscC_in_ag(X2, X3)) memberscC_in_ag([], X1) -> memberscC_out_ag([], X1) U17_ag(X1, X2, X3, memberscC_out_ag(X2, X3)) -> memberscC_out_ag(.(X1, X2), X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) selectcA_in_aga(x1, x2, x3) = selectcA_in_aga(x2) selectcA_out_aga(x1, x2, x3) = selectcA_out_aga(x1, x2, x3) U12_aga(x1, x2, x3, x4, x5) = U12_aga(x2, x3, x5) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) memberscC_in_ag(x1, x2) = memberscC_in_ag(x2) U16_ag(x1, x2, x3, x4) = U16_ag(x3, x4) U17_ag(x1, x2, x3, x4) = U17_ag(x1, x3, x4) memberscC_out_ag(x1, x2) = memberscC_out_ag(x1, x2) MEMBERD_IN_AG(x1, x2) = MEMBERD_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (72) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (73) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERD_IN_AG(X1, .(X2, X3)) -> MEMBERD_IN_AG(X1, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBERD_IN_AG(x1, x2) = MEMBERD_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (74) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERD_IN_AG(.(X2, X3)) -> MEMBERD_IN_AG(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (76) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *MEMBERD_IN_AG(.(X2, X3)) -> MEMBERD_IN_AG(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (77) YES ---------------------------------------- (78) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERSC_IN_AG(.(X1, X2), X3) -> U3_AG(X1, X2, X3, membercD_in_ag(X1, X3)) U3_AG(X1, X2, X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X2, X3) The TRS R consists of the following rules: selectcA_in_aga(X1, .(X1, X2), X2) -> selectcA_out_aga(X1, .(X1, X2), X2) selectcA_in_aga(X1, .(X2, X3), .(X2, X4)) -> U12_aga(X1, X2, X3, X4, selectcA_in_aga(X1, X3, X4)) U12_aga(X1, X2, X3, X4, selectcA_out_aga(X1, X3, X4)) -> selectcA_out_aga(X1, .(X2, X3), .(X2, X4)) membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) memberscC_in_ag(.(X1, X2), X3) -> U16_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U16_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> U17_ag(X1, X2, X3, memberscC_in_ag(X2, X3)) memberscC_in_ag([], X1) -> memberscC_out_ag([], X1) U17_ag(X1, X2, X3, memberscC_out_ag(X2, X3)) -> memberscC_out_ag(.(X1, X2), X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) selectcA_in_aga(x1, x2, x3) = selectcA_in_aga(x2) selectcA_out_aga(x1, x2, x3) = selectcA_out_aga(x1, x2, x3) U12_aga(x1, x2, x3, x4, x5) = U12_aga(x2, x3, x5) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) memberscC_in_ag(x1, x2) = memberscC_in_ag(x2) U16_ag(x1, x2, x3, x4) = U16_ag(x3, x4) U17_ag(x1, x2, x3, x4) = U17_ag(x1, x3, x4) memberscC_out_ag(x1, x2) = memberscC_out_ag(x1, x2) MEMBERSC_IN_AG(x1, x2) = MEMBERSC_IN_AG(x2) U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (80) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERSC_IN_AG(.(X1, X2), X3) -> U3_AG(X1, X2, X3, membercD_in_ag(X1, X3)) U3_AG(X1, X2, X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X2, X3) The TRS R consists of the following rules: membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) MEMBERSC_IN_AG(x1, x2) = MEMBERSC_IN_AG(x2) U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (81) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERSC_IN_AG(X3) -> U3_AG(X3, membercD_in_ag(X3)) U3_AG(X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X3) The TRS R consists of the following rules: membercD_in_ag(.(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(.(X2, X3)) -> U18_ag(X2, X3, membercD_in_ag(X3)) U18_ag(X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercD_in_ag(x0) U18_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (SOUND) By narrowing [LPAR04] the rule MEMBERSC_IN_AG(X3) -> U3_AG(X3, membercD_in_ag(X3)) at position [1] we obtained the following new rules [LPAR04]: (MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), membercD_out_ag(x0, .(x0, x1))),MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), membercD_out_ag(x0, .(x0, x1)))) (MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), U18_ag(x0, x1, membercD_in_ag(x1))),MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), U18_ag(x0, x1, membercD_in_ag(x1)))) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AG(X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X3) MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), membercD_out_ag(x0, .(x0, x1))) MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), U18_ag(x0, x1, membercD_in_ag(x1))) The TRS R consists of the following rules: membercD_in_ag(.(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(.(X2, X3)) -> U18_ag(X2, X3, membercD_in_ag(X3)) U18_ag(X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercD_in_ag(x0) U18_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (85) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AG(X3, membercD_out_ag(X1, X3)) -> MEMBERSC_IN_AG(X3) we obtained the following new rules [LPAR04]: (U3_AG(.(z0, z1), membercD_out_ag(z0, .(z0, z1))) -> MEMBERSC_IN_AG(.(z0, z1)),U3_AG(.(z0, z1), membercD_out_ag(z0, .(z0, z1))) -> MEMBERSC_IN_AG(.(z0, z1))) (U3_AG(.(z0, z1), membercD_out_ag(x1, .(z0, z1))) -> MEMBERSC_IN_AG(.(z0, z1)),U3_AG(.(z0, z1), membercD_out_ag(x1, .(z0, z1))) -> MEMBERSC_IN_AG(.(z0, z1))) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), membercD_out_ag(x0, .(x0, x1))) MEMBERSC_IN_AG(.(x0, x1)) -> U3_AG(.(x0, x1), U18_ag(x0, x1, membercD_in_ag(x1))) U3_AG(.(z0, z1), membercD_out_ag(z0, .(z0, z1))) -> MEMBERSC_IN_AG(.(z0, z1)) U3_AG(.(z0, z1), membercD_out_ag(x1, .(z0, z1))) -> MEMBERSC_IN_AG(.(z0, z1)) The TRS R consists of the following rules: membercD_in_ag(.(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(.(X2, X3)) -> U18_ag(X2, X3, membercD_in_ag(X3)) U18_ag(X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercD_in_ag(x0) U18_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (87) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECTA_IN_AGA(X1, .(X2, X3), .(X2, X4)) -> SELECTA_IN_AGA(X1, X3, X4) The TRS R consists of the following rules: selectcA_in_aga(X1, .(X1, X2), X2) -> selectcA_out_aga(X1, .(X1, X2), X2) selectcA_in_aga(X1, .(X2, X3), .(X2, X4)) -> U12_aga(X1, X2, X3, X4, selectcA_in_aga(X1, X3, X4)) U12_aga(X1, X2, X3, X4, selectcA_out_aga(X1, X3, X4)) -> selectcA_out_aga(X1, .(X2, X3), .(X2, X4)) membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) memberscC_in_ag(.(X1, X2), X3) -> U16_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U16_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> U17_ag(X1, X2, X3, memberscC_in_ag(X2, X3)) memberscC_in_ag([], X1) -> memberscC_out_ag([], X1) U17_ag(X1, X2, X3, memberscC_out_ag(X2, X3)) -> memberscC_out_ag(.(X1, X2), X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) selectcA_in_aga(x1, x2, x3) = selectcA_in_aga(x2) selectcA_out_aga(x1, x2, x3) = selectcA_out_aga(x1, x2, x3) U12_aga(x1, x2, x3, x4, x5) = U12_aga(x2, x3, x5) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) memberscC_in_ag(x1, x2) = memberscC_in_ag(x2) U16_ag(x1, x2, x3, x4) = U16_ag(x3, x4) U17_ag(x1, x2, x3, x4) = U17_ag(x1, x3, x4) memberscC_out_ag(x1, x2) = memberscC_out_ag(x1, x2) SELECTA_IN_AGA(x1, x2, x3) = SELECTA_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (88) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (89) Obligation: Pi DP problem: The TRS P consists of the following rules: SELECTA_IN_AGA(X1, .(X2, X3), .(X2, X4)) -> SELECTA_IN_AGA(X1, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SELECTA_IN_AGA(x1, x2, x3) = SELECTA_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (90) Obligation: Pi DP problem: The TRS P consists of the following rules: COLOR_MAPB_IN_AG(.(region(X1, X2, X3), X4), X5) -> U7_AG(X1, X2, X3, X4, X5, selectcA_in_aga(X2, X5, X6)) U7_AG(X1, X2, X3, X4, X5, selectcA_out_aga(X2, X5, X6)) -> U9_AG(X1, X2, X3, X4, X5, memberscC_in_ag(X3, X6)) U9_AG(X1, X2, X3, X4, X5, memberscC_out_ag(X3, X6)) -> COLOR_MAPB_IN_AG(X4, X5) The TRS R consists of the following rules: selectcA_in_aga(X1, .(X1, X2), X2) -> selectcA_out_aga(X1, .(X1, X2), X2) selectcA_in_aga(X1, .(X2, X3), .(X2, X4)) -> U12_aga(X1, X2, X3, X4, selectcA_in_aga(X1, X3, X4)) U12_aga(X1, X2, X3, X4, selectcA_out_aga(X1, X3, X4)) -> selectcA_out_aga(X1, .(X2, X3), .(X2, X4)) membercD_in_ag(X1, .(X1, X2)) -> membercD_out_ag(X1, .(X1, X2)) membercD_in_ag(X1, .(X2, X3)) -> U18_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U18_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> membercD_out_ag(X1, .(X2, X3)) memberscC_in_ag(.(X1, X2), X3) -> U16_ag(X1, X2, X3, membercD_in_ag(X1, X3)) U16_ag(X1, X2, X3, membercD_out_ag(X1, X3)) -> U17_ag(X1, X2, X3, memberscC_in_ag(X2, X3)) memberscC_in_ag([], X1) -> memberscC_out_ag([], X1) U17_ag(X1, X2, X3, memberscC_out_ag(X2, X3)) -> memberscC_out_ag(.(X1, X2), X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) region(x1, x2, x3) = region(x2, x3) selectcA_in_aga(x1, x2, x3) = selectcA_in_aga(x2) selectcA_out_aga(x1, x2, x3) = selectcA_out_aga(x1, x2, x3) U12_aga(x1, x2, x3, x4, x5) = U12_aga(x2, x3, x5) membercD_in_ag(x1, x2) = membercD_in_ag(x2) membercD_out_ag(x1, x2) = membercD_out_ag(x1, x2) U18_ag(x1, x2, x3, x4) = U18_ag(x2, x3, x4) memberscC_in_ag(x1, x2) = memberscC_in_ag(x2) U16_ag(x1, x2, x3, x4) = U16_ag(x3, x4) U17_ag(x1, x2, x3, x4) = U17_ag(x1, x3, x4) memberscC_out_ag(x1, x2) = memberscC_out_ag(x1, x2) COLOR_MAPB_IN_AG(x1, x2) = COLOR_MAPB_IN_AG(x2) U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x5, x6) U9_AG(x1, x2, x3, x4, x5, x6) = U9_AG(x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (91) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 5, "program": { "directives": [], "clauses": [ [ "(color_map (. Region Regions) Colors)", "(',' (color_region Region Colors) (color_map Regions Colors))" ], [ "(color_map ([]) Colors)", null ], [ "(color_region (region Name Color Neighbors) Colors)", "(',' (select Color Colors Colors1) (members Neighbors Colors1))" ], [ "(select X (. X Xs) Xs)", null ], [ "(select X (. Y Ys) (. Y Zs))", "(select X Ys Zs)" ], [ "(members (. X Xs) Ys)", "(',' (member X Ys) (members Xs Ys))" ], [ "(members ([]) Ys)", null ], [ "(member X (. X X1))", null ], [ "(member X (. X2 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "28": { "goal": [{ "clause": 0, "scope": 1, "term": "(color_map T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "29": { "goal": [{ "clause": 1, "scope": 1, "term": "(color_map T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "type": "Nodes", "293": { "goal": [{ "clause": 2, "scope": 2, "term": "(color_region T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "392": { "goal": [ { "clause": 7, "scope": 5, "term": "(member T95 T94)" }, { "clause": 8, "scope": 5, "term": "(member T95 T94)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "352": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T72 T71 X74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T71"], "free": ["X74"], "exprvars": [] } }, "353": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "430": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "332": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (select T41 T40 X41) (members T42 X41))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "376": { "goal": [ { "clause": 5, "scope": 4, "term": "(members T48 T47)" }, { "clause": 6, "scope": 4, "term": "(members T48 T47)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T47"], "free": [], "exprvars": [] } }, "398": { "goal": [{ "clause": 7, "scope": 5, "term": "(member T95 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "333": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "377": { "goal": [{ "clause": 5, "scope": 4, "term": "(members T48 T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T47"], "free": [], "exprvars": [] } }, "399": { "goal": [{ "clause": 8, "scope": 5, "term": "(member T95 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "378": { "goal": [{ "clause": 6, "scope": 4, "term": "(members T48 T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T47"], "free": [], "exprvars": [] } }, "379": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T95 T94) (members T96 T94))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "336": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T41 T40 X41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "337": { "goal": [{ "clause": -1, "scope": -1, "term": "(members T48 T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T47"], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (color_region T18 T17) (color_map T19 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "31": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "380": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "260": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_region T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "381": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T95 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "383": { "goal": [{ "clause": -1, "scope": -1, "term": "(members T101 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "341": { "goal": [ { "clause": 3, "scope": 3, "term": "(select T41 T40 X41)" }, { "clause": 4, "scope": 3, "term": "(select T41 T40 X41)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "342": { "goal": [{ "clause": 3, "scope": 3, "term": "(select T41 T40 X41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "266": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_map T24 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "343": { "goal": [{ "clause": 4, "scope": 3, "term": "(select T41 T40 X41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "420": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T125 T124)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T124"], "free": [], "exprvars": [] } }, "422": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_map T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "401": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(color_map T1 T2)" }, { "clause": 1, "scope": 1, "term": "(color_map T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "347": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "348": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "403": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "425": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "349": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "426": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "405": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "427": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "428": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "429": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 5, "to": 6, "label": "CASE" }, { "from": 6, "to": 28, "label": "PARALLEL" }, { "from": 6, "to": 29, "label": "PARALLEL" }, { "from": 28, "to": 30, "label": "EVAL with clause\ncolor_map(.(X15, X16), X17) :- ','(color_region(X15, X17), color_map(X16, X17)).\nand substitutionX15 -> T18,\nX16 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX17 -> T17,\nT15 -> T18,\nT16 -> T19" }, { "from": 28, "to": 31, "label": "EVAL-BACKTRACK" }, { "from": 29, "to": 428, "label": "EVAL with clause\ncolor_map([], X139).\nand substitutionT1 -> [],\nT2 -> T141,\nX139 -> T141" }, { "from": 29, "to": 429, "label": "EVAL-BACKTRACK" }, { "from": 30, "to": 260, "label": "SPLIT 1" }, { "from": 30, "to": 266, "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nreplacements:T19 -> T24" }, { "from": 260, "to": 293, "label": "CASE" }, { "from": 266, "to": 5, "label": "INSTANCE with matching:\nT1 -> T24\nT2 -> T17" }, { "from": 293, "to": 332, "label": "EVAL with clause\ncolor_region(region(X37, X38, X39), X40) :- ','(select(X38, X40, X41), members(X39, X41)).\nand substitutionX37 -> T37,\nX38 -> T41,\nX39 -> T42,\nT18 -> region(T37, T41, T42),\nT17 -> T40,\nX40 -> T40,\nT38 -> T41,\nT39 -> T42" }, { "from": 293, "to": 333, "label": "EVAL-BACKTRACK" }, { "from": 332, "to": 336, "label": "SPLIT 1" }, { "from": 332, "to": 337, "label": "SPLIT 2\nnew knowledge:\nT41 is ground\nT40 is ground\nT47 is ground\nreplacements:X41 -> T47,\nT42 -> T48" }, { "from": 336, "to": 341, "label": "CASE" }, { "from": 337, "to": 376, "label": "CASE" }, { "from": 341, "to": 342, "label": "PARALLEL" }, { "from": 341, "to": 343, "label": "PARALLEL" }, { "from": 342, "to": 347, "label": "EVAL with clause\nselect(X58, .(X58, X59), X59).\nand substitutionT41 -> T61,\nX58 -> T61,\nX59 -> T62,\nT40 -> .(T61, T62),\nX41 -> T62" }, { "from": 342, "to": 348, "label": "EVAL-BACKTRACK" }, { "from": 343, "to": 352, "label": "EVAL with clause\nselect(X70, .(X71, X72), .(X71, X73)) :- select(X70, X72, X73).\nand substitutionT41 -> T72,\nX70 -> T72,\nX71 -> T70,\nX72 -> T71,\nT40 -> .(T70, T71),\nX73 -> X74,\nX41 -> .(T70, X74),\nT69 -> T72" }, { "from": 343, "to": 353, "label": "EVAL-BACKTRACK" }, { "from": 347, "to": 349, "label": "SUCCESS" }, { "from": 352, "to": 336, "label": "INSTANCE with matching:\nT41 -> T72\nT40 -> T71\nX41 -> X74" }, { "from": 376, "to": 377, "label": "PARALLEL" }, { "from": 376, "to": 378, "label": "PARALLEL" }, { "from": 377, "to": 379, "label": "EVAL with clause\nmembers(.(X94, X95), X96) :- ','(member(X94, X96), members(X95, X96)).\nand substitutionX94 -> T95,\nX95 -> T96,\nT48 -> .(T95, T96),\nT47 -> T94,\nX96 -> T94,\nT92 -> T95,\nT93 -> T96" }, { "from": 377, "to": 380, "label": "EVAL-BACKTRACK" }, { "from": 378, "to": 425, "label": "EVAL with clause\nmembers([], X133).\nand substitutionT48 -> [],\nT47 -> T135,\nX133 -> T135" }, { "from": 378, "to": 426, "label": "EVAL-BACKTRACK" }, { "from": 379, "to": 381, "label": "SPLIT 1" }, { "from": 379, "to": 383, "label": "SPLIT 2\nnew knowledge:\nT95 is ground\nT94 is ground\nreplacements:T96 -> T101" }, { "from": 381, "to": 392, "label": "CASE" }, { "from": 383, "to": 337, "label": "INSTANCE with matching:\nT48 -> T101\nT47 -> T94" }, { "from": 392, "to": 398, "label": "PARALLEL" }, { "from": 392, "to": 399, "label": "PARALLEL" }, { "from": 398, "to": 401, "label": "EVAL with clause\nmember(X113, .(X113, X114)).\nand substitutionT95 -> T114,\nX113 -> T114,\nX114 -> T115,\nT94 -> .(T114, T115)" }, { "from": 398, "to": 403, "label": "EVAL-BACKTRACK" }, { "from": 399, "to": 420, "label": "EVAL with clause\nmember(X121, .(X122, X123)) :- member(X121, X123).\nand substitutionT95 -> T125,\nX121 -> T125,\nX122 -> T123,\nX123 -> T124,\nT94 -> .(T123, T124),\nT122 -> T125" }, { "from": 399, "to": 422, "label": "EVAL-BACKTRACK" }, { "from": 401, "to": 405, "label": "SUCCESS" }, { "from": 420, "to": 381, "label": "INSTANCE with matching:\nT95 -> T125\nT94 -> T124" }, { "from": 425, "to": 427, "label": "SUCCESS" }, { "from": 428, "to": 430, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (92) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f5_in(T17) -> U1(f30_in(T17), T17) U1(f30_out1, T17) -> f5_out1 f5_in(T141) -> f5_out1 f336_in(.(T61, T62)) -> f336_out1(T61, T62) f336_in(.(T70, T71)) -> U2(f336_in(T71), .(T70, T71)) U2(f336_out1(T72, X74), .(T70, T71)) -> f336_out1(T72, .(T70, X74)) f337_in(T94) -> U3(f379_in(T94), T94) U3(f379_out1(T95, T96), T94) -> f337_out1(.(T95, T96)) f337_in(T135) -> f337_out1([]) f381_in(.(T114, T115)) -> f381_out1(T114) f381_in(.(T123, T124)) -> U4(f381_in(T124), .(T123, T124)) U4(f381_out1(T125), .(T123, T124)) -> f381_out1(T125) f260_in(T40) -> U5(f332_in(T40), T40) U5(f332_out1(T41, X41, T42), T40) -> f260_out1 f30_in(T17) -> U6(f260_in(T17), T17) U6(f260_out1, T17) -> U7(f5_in(T17), T17) U7(f5_out1, T17) -> f30_out1 f332_in(T40) -> U8(f336_in(T40), T40) U8(f336_out1(T41, T47), T40) -> U9(f337_in(T47), T40, T41, T47) U9(f337_out1(T48), T40, T41, T47) -> f332_out1(T41, T47, T48) f379_in(T94) -> U10(f381_in(T94), T94) U10(f381_out1(T95), T94) -> U11(f337_in(T94), T94, T95) U11(f337_out1(T101), T94, T95) -> f379_out1(T95, T101) Q is empty. ---------------------------------------- (93) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: F5_IN(T17) -> U1^1(f30_in(T17), T17) F5_IN(T17) -> F30_IN(T17) F336_IN(.(T70, T71)) -> U2^1(f336_in(T71), .(T70, T71)) F336_IN(.(T70, T71)) -> F336_IN(T71) F337_IN(T94) -> U3^1(f379_in(T94), T94) F337_IN(T94) -> F379_IN(T94) F381_IN(.(T123, T124)) -> U4^1(f381_in(T124), .(T123, T124)) F381_IN(.(T123, T124)) -> F381_IN(T124) F260_IN(T40) -> U5^1(f332_in(T40), T40) F260_IN(T40) -> F332_IN(T40) F30_IN(T17) -> U6^1(f260_in(T17), T17) F30_IN(T17) -> F260_IN(T17) U6^1(f260_out1, T17) -> U7^1(f5_in(T17), T17) U6^1(f260_out1, T17) -> F5_IN(T17) F332_IN(T40) -> U8^1(f336_in(T40), T40) F332_IN(T40) -> F336_IN(T40) U8^1(f336_out1(T41, T47), T40) -> U9^1(f337_in(T47), T40, T41, T47) U8^1(f336_out1(T41, T47), T40) -> F337_IN(T47) F379_IN(T94) -> U10^1(f381_in(T94), T94) F379_IN(T94) -> F381_IN(T94) U10^1(f381_out1(T95), T94) -> U11^1(f337_in(T94), T94, T95) U10^1(f381_out1(T95), T94) -> F337_IN(T94) The TRS R consists of the following rules: f5_in(T17) -> U1(f30_in(T17), T17) U1(f30_out1, T17) -> f5_out1 f5_in(T141) -> f5_out1 f336_in(.(T61, T62)) -> f336_out1(T61, T62) f336_in(.(T70, T71)) -> U2(f336_in(T71), .(T70, T71)) U2(f336_out1(T72, X74), .(T70, T71)) -> f336_out1(T72, .(T70, X74)) f337_in(T94) -> U3(f379_in(T94), T94) U3(f379_out1(T95, T96), T94) -> f337_out1(.(T95, T96)) f337_in(T135) -> f337_out1([]) f381_in(.(T114, T115)) -> f381_out1(T114) f381_in(.(T123, T124)) -> U4(f381_in(T124), .(T123, T124)) U4(f381_out1(T125), .(T123, T124)) -> f381_out1(T125) f260_in(T40) -> U5(f332_in(T40), T40) U5(f332_out1(T41, X41, T42), T40) -> f260_out1 f30_in(T17) -> U6(f260_in(T17), T17) U6(f260_out1, T17) -> U7(f5_in(T17), T17) U7(f5_out1, T17) -> f30_out1 f332_in(T40) -> U8(f336_in(T40), T40) U8(f336_out1(T41, T47), T40) -> U9(f337_in(T47), T40, T41, T47) U9(f337_out1(T48), T40, T41, T47) -> f332_out1(T41, T47, T48) f379_in(T94) -> U10(f381_in(T94), T94) U10(f381_out1(T95), T94) -> U11(f337_in(T94), T94, T95) U11(f337_out1(T101), T94, T95) -> f379_out1(T95, T101) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 14 less nodes. ---------------------------------------- (96) Complex Obligation (AND) ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: F381_IN(.(T123, T124)) -> F381_IN(T124) The TRS R consists of the following rules: f5_in(T17) -> U1(f30_in(T17), T17) U1(f30_out1, T17) -> f5_out1 f5_in(T141) -> f5_out1 f336_in(.(T61, T62)) -> f336_out1(T61, T62) f336_in(.(T70, T71)) -> U2(f336_in(T71), .(T70, T71)) U2(f336_out1(T72, X74), .(T70, T71)) -> f336_out1(T72, .(T70, X74)) f337_in(T94) -> U3(f379_in(T94), T94) U3(f379_out1(T95, T96), T94) -> f337_out1(.(T95, T96)) f337_in(T135) -> f337_out1([]) f381_in(.(T114, T115)) -> f381_out1(T114) f381_in(.(T123, T124)) -> U4(f381_in(T124), .(T123, T124)) U4(f381_out1(T125), .(T123, T124)) -> f381_out1(T125) f260_in(T40) -> U5(f332_in(T40), T40) U5(f332_out1(T41, X41, T42), T40) -> f260_out1 f30_in(T17) -> U6(f260_in(T17), T17) U6(f260_out1, T17) -> U7(f5_in(T17), T17) U7(f5_out1, T17) -> f30_out1 f332_in(T40) -> U8(f336_in(T40), T40) U8(f336_out1(T41, T47), T40) -> U9(f337_in(T47), T40, T41, T47) U9(f337_out1(T48), T40, T41, T47) -> f332_out1(T41, T47, T48) f379_in(T94) -> U10(f381_in(T94), T94) U10(f381_out1(T95), T94) -> U11(f337_in(T94), T94, T95) U11(f337_out1(T101), T94, T95) -> f379_out1(T95, T101) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: F381_IN(.(T123, T124)) -> F381_IN(T124) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *F381_IN(.(T123, T124)) -> F381_IN(T124) The graph contains the following edges 1 > 1 ---------------------------------------- (101) YES ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: F337_IN(T94) -> F379_IN(T94) F379_IN(T94) -> U10^1(f381_in(T94), T94) U10^1(f381_out1(T95), T94) -> F337_IN(T94) The TRS R consists of the following rules: f5_in(T17) -> U1(f30_in(T17), T17) U1(f30_out1, T17) -> f5_out1 f5_in(T141) -> f5_out1 f336_in(.(T61, T62)) -> f336_out1(T61, T62) f336_in(.(T70, T71)) -> U2(f336_in(T71), .(T70, T71)) U2(f336_out1(T72, X74), .(T70, T71)) -> f336_out1(T72, .(T70, X74)) f337_in(T94) -> U3(f379_in(T94), T94) U3(f379_out1(T95, T96), T94) -> f337_out1(.(T95, T96)) f337_in(T135) -> f337_out1([]) f381_in(.(T114, T115)) -> f381_out1(T114) f381_in(.(T123, T124)) -> U4(f381_in(T124), .(T123, T124)) U4(f381_out1(T125), .(T123, T124)) -> f381_out1(T125) f260_in(T40) -> U5(f332_in(T40), T40) U5(f332_out1(T41, X41, T42), T40) -> f260_out1 f30_in(T17) -> U6(f260_in(T17), T17) U6(f260_out1, T17) -> U7(f5_in(T17), T17) U7(f5_out1, T17) -> f30_out1 f332_in(T40) -> U8(f336_in(T40), T40) U8(f336_out1(T41, T47), T40) -> U9(f337_in(T47), T40, T41, T47) U9(f337_out1(T48), T40, T41, T47) -> f332_out1(T41, T47, T48) f379_in(T94) -> U10(f381_in(T94), T94) U10(f381_out1(T95), T94) -> U11(f337_in(T94), T94, T95) U11(f337_out1(T101), T94, T95) -> f379_out1(T95, T101) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = F379_IN(.(T114, T115)) evaluates to t =F379_IN(.(T114, T115)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F379_IN(.(T114, T115)) -> U10^1(f381_in(.(T114, T115)), .(T114, T115)) with rule F379_IN(T94) -> U10^1(f381_in(T94), T94) at position [] and matcher [T94 / .(T114, T115)] U10^1(f381_in(.(T114, T115)), .(T114, T115)) -> U10^1(f381_out1(T114), .(T114, T115)) with rule f381_in(.(T114', T115')) -> f381_out1(T114') at position [0] and matcher [T114' / T114, T115' / T115] U10^1(f381_out1(T114), .(T114, T115)) -> F337_IN(.(T114, T115)) with rule U10^1(f381_out1(T95), T94') -> F337_IN(T94') at position [] and matcher [T95 / T114, T94' / .(T114, T115)] F337_IN(.(T114, T115)) -> F379_IN(.(T114, T115)) with rule F337_IN(T94) -> F379_IN(T94) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (104) NO ---------------------------------------- (105) Obligation: Q DP problem: The TRS P consists of the following rules: F336_IN(.(T70, T71)) -> F336_IN(T71) The TRS R consists of the following rules: f5_in(T17) -> U1(f30_in(T17), T17) U1(f30_out1, T17) -> f5_out1 f5_in(T141) -> f5_out1 f336_in(.(T61, T62)) -> f336_out1(T61, T62) f336_in(.(T70, T71)) -> U2(f336_in(T71), .(T70, T71)) U2(f336_out1(T72, X74), .(T70, T71)) -> f336_out1(T72, .(T70, X74)) f337_in(T94) -> U3(f379_in(T94), T94) U3(f379_out1(T95, T96), T94) -> f337_out1(.(T95, T96)) f337_in(T135) -> f337_out1([]) f381_in(.(T114, T115)) -> f381_out1(T114) f381_in(.(T123, T124)) -> U4(f381_in(T124), .(T123, T124)) U4(f381_out1(T125), .(T123, T124)) -> f381_out1(T125) f260_in(T40) -> U5(f332_in(T40), T40) U5(f332_out1(T41, X41, T42), T40) -> f260_out1 f30_in(T17) -> U6(f260_in(T17), T17) U6(f260_out1, T17) -> U7(f5_in(T17), T17) U7(f5_out1, T17) -> f30_out1 f332_in(T40) -> U8(f336_in(T40), T40) U8(f336_out1(T41, T47), T40) -> U9(f337_in(T47), T40, T41, T47) U9(f337_out1(T48), T40, T41, T47) -> f332_out1(T41, T47, T48) f379_in(T94) -> U10(f381_in(T94), T94) U10(f381_out1(T95), T94) -> U11(f337_in(T94), T94, T95) U11(f337_out1(T101), T94, T95) -> f379_out1(T95, T101) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (106) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: F336_IN(.(T70, T71)) -> F336_IN(T71) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (108) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *F336_IN(.(T70, T71)) -> F336_IN(T71) The graph contains the following edges 1 > 1 ---------------------------------------- (109) YES ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: F5_IN(T17) -> F30_IN(T17) F30_IN(T17) -> U6^1(f260_in(T17), T17) U6^1(f260_out1, T17) -> F5_IN(T17) The TRS R consists of the following rules: f5_in(T17) -> U1(f30_in(T17), T17) U1(f30_out1, T17) -> f5_out1 f5_in(T141) -> f5_out1 f336_in(.(T61, T62)) -> f336_out1(T61, T62) f336_in(.(T70, T71)) -> U2(f336_in(T71), .(T70, T71)) U2(f336_out1(T72, X74), .(T70, T71)) -> f336_out1(T72, .(T70, X74)) f337_in(T94) -> U3(f379_in(T94), T94) U3(f379_out1(T95, T96), T94) -> f337_out1(.(T95, T96)) f337_in(T135) -> f337_out1([]) f381_in(.(T114, T115)) -> f381_out1(T114) f381_in(.(T123, T124)) -> U4(f381_in(T124), .(T123, T124)) U4(f381_out1(T125), .(T123, T124)) -> f381_out1(T125) f260_in(T40) -> U5(f332_in(T40), T40) U5(f332_out1(T41, X41, T42), T40) -> f260_out1 f30_in(T17) -> U6(f260_in(T17), T17) U6(f260_out1, T17) -> U7(f5_in(T17), T17) U7(f5_out1, T17) -> f30_out1 f332_in(T40) -> U8(f336_in(T40), T40) U8(f336_out1(T41, T47), T40) -> U9(f337_in(T47), T40, T41, T47) U9(f337_out1(T48), T40, T41, T47) -> f332_out1(T41, T47, T48) f379_in(T94) -> U10(f381_in(T94), T94) U10(f381_out1(T95), T94) -> U11(f337_in(T94), T94, T95) U11(f337_out1(T101), T94, T95) -> f379_out1(T95, T101) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = F30_IN(.(T61, T62)) evaluates to t =F30_IN(.(T61, T62)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F30_IN(.(T61, T62)) -> U6^1(f260_in(.(T61, T62)), .(T61, T62)) with rule F30_IN(T17) -> U6^1(f260_in(T17), T17) at position [] and matcher [T17 / .(T61, T62)] U6^1(f260_in(.(T61, T62)), .(T61, T62)) -> U6^1(U5(f332_in(.(T61, T62)), .(T61, T62)), .(T61, T62)) with rule f260_in(T40') -> U5(f332_in(T40'), T40') at position [0] and matcher [T40' / .(T61, T62)] U6^1(U5(f332_in(.(T61, T62)), .(T61, T62)), .(T61, T62)) -> U6^1(U5(U8(f336_in(.(T61, T62)), .(T61, T62)), .(T61, T62)), .(T61, T62)) with rule f332_in(T40') -> U8(f336_in(T40'), T40') at position [0,0] and matcher [T40' / .(T61, T62)] U6^1(U5(U8(f336_in(.(T61, T62)), .(T61, T62)), .(T61, T62)), .(T61, T62)) -> U6^1(U5(U8(f336_out1(T61, T62), .(T61, T62)), .(T61, T62)), .(T61, T62)) with rule f336_in(.(T61', T62')) -> f336_out1(T61', T62') at position [0,0,0] and matcher [T61' / T61, T62' / T62] U6^1(U5(U8(f336_out1(T61, T62), .(T61, T62)), .(T61, T62)), .(T61, T62)) -> U6^1(U5(U9(f337_in(T62), .(T61, T62), T61, T62), .(T61, T62)), .(T61, T62)) with rule U8(f336_out1(T41, T47'), T40'') -> U9(f337_in(T47'), T40'', T41, T47') at position [0,0] and matcher [T41 / T61, T47' / T62, T40'' / .(T61, T62)] U6^1(U5(U9(f337_in(T62), .(T61, T62), T61, T62), .(T61, T62)), .(T61, T62)) -> U6^1(U5(U9(f337_out1([]), .(T61, T62), T61, T62), .(T61, T62)), .(T61, T62)) with rule f337_in(T135) -> f337_out1([]) at position [0,0,0] and matcher [T135 / T62] U6^1(U5(U9(f337_out1([]), .(T61, T62), T61, T62), .(T61, T62)), .(T61, T62)) -> U6^1(U5(f332_out1(T61, T62, []), .(T61, T62)), .(T61, T62)) with rule U9(f337_out1(T48), T40', T41', T47) -> f332_out1(T41', T47, T48) at position [0,0] and matcher [T48 / [], T40' / .(T61, T62), T41' / T61, T47 / T62] U6^1(U5(f332_out1(T61, T62, []), .(T61, T62)), .(T61, T62)) -> U6^1(f260_out1, .(T61, T62)) with rule U5(f332_out1(T41, X41, T42), T40) -> f260_out1 at position [0] and matcher [T41 / T61, X41 / T62, T42 / [], T40 / .(T61, T62)] U6^1(f260_out1, .(T61, T62)) -> F5_IN(.(T61, T62)) with rule U6^1(f260_out1, T17') -> F5_IN(T17') at position [] and matcher [T17' / .(T61, T62)] F5_IN(.(T61, T62)) -> F30_IN(.(T61, T62)) with rule F5_IN(T17) -> F30_IN(T17) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (112) NO ---------------------------------------- (113) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(color_map (. Region Regions) Colors)", "(',' (color_region Region Colors) (color_map Regions Colors))" ], [ "(color_map ([]) Colors)", null ], [ "(color_region (region Name Color Neighbors) Colors)", "(',' (select Color Colors Colors1) (members Neighbors Colors1))" ], [ "(select X (. X Xs) Xs)", null ], [ "(select X (. Y Ys) (. Y Zs))", "(select X Ys Zs)" ], [ "(members (. X Xs) Ys)", "(',' (member X Ys) (members Xs Ys))" ], [ "(members ([]) Ys)", null ], [ "(member X (. X X1))", null ], [ "(member X (. 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"relations": [] }, "ground": ["T71"], "free": ["X74"], "exprvars": [] } }, "374": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T95 T94) (members T96 T94))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "375": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "431": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T95 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], "exprvars": [] } }, "432": { "goal": [{ "clause": -1, "scope": -1, "term": "(members T101 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T94"], "free": [], 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"relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "40": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_region T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "62": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [{ "clause": -1, "scope": -1, "term": "(color_map T24 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "63": { "goal": [{ "clause": -1, "scope": -1, "term": "(select T41 T40 X41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } }, "42": { "goal": [{ "clause": 2, "scope": 2, "term": "(color_region T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "64": { "goal": [{ "clause": -1, "scope": -1, "term": "(members T48 T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T47"], "free": [], "exprvars": [] } }, "65": { "goal": [ { "clause": 3, "scope": 3, "term": "(select T41 T40 X41)" }, { "clause": 4, "scope": 3, "term": "(select T41 T40 X41)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T40"], "free": ["X41"], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 34, "label": "PARALLEL" }, { "from": 4, "to": 35, "label": "PARALLEL" }, { "from": 34, "to": 36, "label": "EVAL with clause\ncolor_map(.(X15, X16), X17) :- ','(color_region(X15, X17), color_map(X16, X17)).\nand substitutionX15 -> T18,\nX16 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX17 -> T17,\nT15 -> T18,\nT16 -> T19" }, { "from": 34, "to": 37, "label": "EVAL-BACKTRACK" }, { "from": 35, "to": 446, "label": "EVAL with clause\ncolor_map([], X139).\nand substitutionT1 -> [],\nT2 -> T141,\nX139 -> T141" }, { "from": 35, "to": 447, "label": "EVAL-BACKTRACK" }, { "from": 36, "to": 40, "label": "SPLIT 1" }, { "from": 36, "to": 41, "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nreplacements:T19 -> T24" }, { "from": 40, "to": 42, "label": "CASE" }, { "from": 41, "to": 3, "label": "INSTANCE with matching:\nT1 -> T24\nT2 -> T17" }, { "from": 42, "to": 61, "label": "EVAL with clause\ncolor_region(region(X37, X38, X39), X40) :- ','(select(X38, X40, X41), members(X39, X41)).\nand substitutionX37 -> T37,\nX38 -> T41,\nX39 -> T42,\nT18 -> region(T37, T41, T42),\nT17 -> T40,\nX40 -> T40,\nT38 -> T41,\nT39 -> T42" }, { "from": 42, "to": 62, "label": "EVAL-BACKTRACK" }, { "from": 61, "to": 63, "label": "SPLIT 1" }, { "from": 61, "to": 64, "label": "SPLIT 2\nnew knowledge:\nT41 is ground\nT40 is ground\nT47 is ground\nreplacements:X41 -> T47,\nT42 -> T48" }, { "from": 63, "to": 65, "label": "CASE" }, { "from": 64, "to": 368, "label": "CASE" }, { "from": 65, "to": 66, "label": "PARALLEL" }, { "from": 65, "to": 67, "label": "PARALLEL" }, { "from": 66, "to": 81, "label": "EVAL with clause\nselect(X58, .(X58, X59), X59).\nand substitutionT41 -> T61,\nX58 -> T61,\nX59 -> T62,\nT40 -> .(T61, T62),\nX41 -> T62" }, { "from": 66, "to": 90, "label": "EVAL-BACKTRACK" }, { "from": 67, "to": 198, "label": "EVAL with clause\nselect(X70, .(X71, X72), .(X71, X73)) :- select(X70, X72, X73).\nand substitutionT41 -> T72,\nX70 -> T72,\nX71 -> T70,\nX72 -> T71,\nT40 -> .(T70, T71),\nX73 -> X74,\nX41 -> .(T70, X74),\nT69 -> T72" }, { "from": 67, "to": 200, "label": "EVAL-BACKTRACK" }, { "from": 81, "to": 96, "label": "SUCCESS" }, { "from": 198, "to": 63, "label": "INSTANCE with matching:\nT41 -> T72\nT40 -> T71\nX41 -> X74" }, { "from": 368, "to": 369, "label": "PARALLEL" }, { "from": 368, "to": 370, "label": "PARALLEL" }, { "from": 369, "to": 374, "label": "EVAL with clause\nmembers(.(X94, X95), X96) :- ','(member(X94, X96), members(X95, X96)).\nand substitutionX94 -> T95,\nX95 -> T96,\nT48 -> .(T95, T96),\nT47 -> T94,\nX96 -> T94,\nT92 -> T95,\nT93 -> T96" }, { "from": 369, "to": 375, "label": "EVAL-BACKTRACK" }, { "from": 370, "to": 441, "label": "EVAL with clause\nmembers([], X133).\nand substitutionT48 -> [],\nT47 -> T135,\nX133 -> T135" }, { "from": 370, "to": 442, "label": "EVAL-BACKTRACK" }, { "from": 374, "to": 431, "label": "SPLIT 1" }, { "from": 374, "to": 432, "label": "SPLIT 2\nnew knowledge:\nT95 is ground\nT94 is ground\nreplacements:T96 -> T101" }, { "from": 431, "to": 433, "label": "CASE" }, { "from": 432, "to": 64, "label": "INSTANCE with matching:\nT48 -> T101\nT47 -> T94" }, { "from": 433, "to": 434, "label": "PARALLEL" }, { "from": 433, "to": 435, "label": "PARALLEL" }, { "from": 434, "to": 436, "label": "EVAL with clause\nmember(X113, .(X113, X114)).\nand substitutionT95 -> T114,\nX113 -> T114,\nX114 -> T115,\nT94 -> .(T114, T115)" }, { "from": 434, "to": 437, "label": "EVAL-BACKTRACK" }, { "from": 435, "to": 439, "label": "EVAL with clause\nmember(X121, .(X122, X123)) :- member(X121, X123).\nand substitutionT95 -> T125,\nX121 -> T125,\nX122 -> T123,\nX123 -> T124,\nT94 -> .(T123, T124),\nT122 -> T125" }, { "from": 435, "to": 440, "label": "EVAL-BACKTRACK" }, { "from": 436, "to": 438, "label": "SUCCESS" }, { "from": 439, "to": 431, "label": "INSTANCE with matching:\nT95 -> T125\nT94 -> T124" }, { "from": 441, "to": 443, "label": "SUCCESS" }, { "from": 446, "to": 457, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (114) Complex Obligation (AND) ---------------------------------------- (115) Obligation: Rules: f439_out(T124) -> f435_out(.(T123, T124)) :|: TRUE f435_in(.(x, x1)) -> f439_in(x1) :|: TRUE f440_out -> f435_out(T94) :|: TRUE f435_in(x2) -> f440_in :|: TRUE f433_in(x3) -> f434_in(x3) :|: TRUE f433_in(x4) -> f435_in(x4) :|: TRUE f434_out(x5) -> f433_out(x5) :|: TRUE f435_out(x6) -> f433_out(x6) :|: TRUE f431_in(x7) -> f433_in(x7) :|: TRUE f433_out(x8) -> f431_out(x8) :|: TRUE f439_in(x9) -> f431_in(x9) :|: TRUE f431_out(x10) -> f439_out(x10) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x11) -> f4_in(x11) :|: TRUE f34_out(x12) -> f4_out(x12) :|: TRUE f35_out(x13) -> f4_out(x13) :|: TRUE f4_in(x14) -> f35_in(x14) :|: TRUE f4_in(x15) -> f34_in(x15) :|: TRUE f34_in(x16) -> f37_in :|: TRUE f34_in(T17) -> f36_in(T17) :|: TRUE f37_out -> f34_out(x17) :|: TRUE f36_out(x18) -> f34_out(x18) :|: TRUE f41_out(x19) -> f36_out(x19) :|: TRUE f40_out(x20) -> f41_in(x20) :|: TRUE f36_in(x21) -> f40_in(x21) :|: TRUE f42_out(x22) -> f40_out(x22) :|: TRUE f40_in(x23) -> f42_in(x23) :|: TRUE f42_in(T40) -> f61_in(T40) :|: TRUE f62_out -> f42_out(x24) :|: TRUE f42_in(x25) -> f62_in :|: TRUE f61_out(x26) -> f42_out(x26) :|: TRUE f61_in(x27) -> f63_in(x27) :|: TRUE f63_out(x28) -> f64_in(x29) :|: TRUE f64_out(x30) -> f61_out(x31) :|: TRUE f368_out(T47) -> f64_out(T47) :|: TRUE f64_in(x32) -> f368_in(x32) :|: TRUE f368_in(x33) -> f370_in(x33) :|: TRUE f369_out(x34) -> f368_out(x34) :|: TRUE f370_out(x35) -> f368_out(x35) :|: TRUE f368_in(x36) -> f369_in(x36) :|: TRUE f369_in(x37) -> f375_in :|: TRUE f369_in(x38) -> f374_in(x38) :|: TRUE f375_out -> f369_out(x39) :|: TRUE f374_out(x40) -> f369_out(x40) :|: TRUE f374_in(x41) -> f431_in(x41) :|: TRUE f431_out(x42) -> f432_in(x42) :|: TRUE f432_out(x43) -> f374_out(x43) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (116) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (117) TRUE ---------------------------------------- (118) Obligation: Rules: f368_out(T47) -> f64_out(T47) :|: TRUE f64_in(x) -> f368_in(x) :|: TRUE f368_in(x1) -> f370_in(x1) :|: TRUE f369_out(x2) -> f368_out(x2) :|: TRUE f370_out(x3) -> f368_out(x3) :|: TRUE f368_in(x4) -> f369_in(x4) :|: TRUE f369_in(x5) -> f375_in :|: TRUE f369_in(T94) -> f374_in(T94) :|: TRUE f375_out -> f369_out(x6) :|: TRUE f374_out(x7) -> f369_out(x7) :|: TRUE f433_in(x8) -> f434_in(x8) :|: TRUE f433_in(x9) -> f435_in(x9) :|: TRUE f434_out(x10) -> f433_out(x10) :|: TRUE f435_out(x11) -> f433_out(x11) :|: TRUE f434_in(x12) -> f437_in :|: TRUE f434_in(.(T114, T115)) -> f436_in :|: TRUE f437_out -> f434_out(x13) :|: TRUE f436_out -> f434_out(.(x14, x15)) :|: TRUE f439_in(T124) -> f431_in(T124) :|: TRUE f431_out(x16) -> f439_out(x16) :|: TRUE f64_out(x17) -> f432_out(x17) :|: TRUE f432_in(x18) -> f64_in(x18) :|: TRUE f374_in(x19) -> f431_in(x19) :|: TRUE f431_out(x20) -> f432_in(x20) :|: TRUE f432_out(x21) -> f374_out(x21) :|: TRUE f436_in -> f436_out :|: TRUE f439_out(x22) -> f435_out(.(x23, x22)) :|: TRUE f435_in(.(x24, x25)) -> f439_in(x25) :|: TRUE f440_out -> f435_out(x26) :|: TRUE f435_in(x27) -> f440_in :|: TRUE f431_in(x28) -> f433_in(x28) :|: TRUE f433_out(x29) -> f431_out(x29) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x30) -> f4_in(x30) :|: TRUE f34_out(x31) -> f4_out(x31) :|: TRUE f35_out(x32) -> f4_out(x32) :|: TRUE f4_in(x33) -> f35_in(x33) :|: TRUE f4_in(x34) -> f34_in(x34) :|: TRUE f34_in(x35) -> f37_in :|: TRUE f34_in(T17) -> f36_in(T17) :|: TRUE f37_out -> f34_out(x36) :|: TRUE f36_out(x37) -> f34_out(x37) :|: TRUE f41_out(x38) -> f36_out(x38) :|: TRUE f40_out(x39) -> f41_in(x39) :|: TRUE f36_in(x40) -> f40_in(x40) :|: TRUE f42_out(x41) -> f40_out(x41) :|: TRUE f40_in(x42) -> f42_in(x42) :|: TRUE f42_in(T40) -> f61_in(T40) :|: TRUE f62_out -> f42_out(x43) :|: TRUE f42_in(x44) -> f62_in :|: TRUE f61_out(x45) -> f42_out(x45) :|: TRUE f61_in(x46) -> f63_in(x46) :|: TRUE f63_out(x47) -> f64_in(x48) :|: TRUE f64_out(x49) -> f61_out(x50) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (119) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (120) TRUE ---------------------------------------- (121) Obligation: Rules: f200_out -> f67_out(T40) :|: TRUE f198_out(T71) -> f67_out(.(T70, T71)) :|: TRUE f67_in(.(x, x1)) -> f198_in(x1) :|: TRUE f67_in(x2) -> f200_in :|: TRUE f65_in(x3) -> f67_in(x3) :|: TRUE f67_out(x4) -> f65_out(x4) :|: TRUE f65_in(x5) -> f66_in(x5) :|: TRUE f66_out(x6) -> f65_out(x6) :|: TRUE f65_out(x7) -> f63_out(x7) :|: TRUE f63_in(x8) -> f65_in(x8) :|: TRUE f63_out(x9) -> f198_out(x9) :|: TRUE f198_in(x10) -> f63_in(x10) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x11) -> f4_in(x11) :|: TRUE f34_out(x12) -> f4_out(x12) :|: TRUE f35_out(x13) -> f4_out(x13) :|: TRUE f4_in(x14) -> f35_in(x14) :|: TRUE f4_in(x15) -> f34_in(x15) :|: TRUE f34_in(x16) -> f37_in :|: TRUE f34_in(T17) -> f36_in(T17) :|: TRUE f37_out -> f34_out(x17) :|: TRUE f36_out(x18) -> f34_out(x18) :|: TRUE f41_out(x19) -> f36_out(x19) :|: TRUE f40_out(x20) -> f41_in(x20) :|: TRUE f36_in(x21) -> f40_in(x21) :|: TRUE f42_out(x22) -> f40_out(x22) :|: TRUE f40_in(x23) -> f42_in(x23) :|: TRUE f42_in(x24) -> f61_in(x24) :|: TRUE f62_out -> f42_out(x25) :|: TRUE f42_in(x26) -> f62_in :|: TRUE f61_out(x27) -> f42_out(x27) :|: TRUE f61_in(x28) -> f63_in(x28) :|: TRUE f63_out(x29) -> f64_in(x30) :|: TRUE f64_out(x31) -> f61_out(x32) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (122) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f67_in(.(x, x1)) -> f198_in(x1) :|: TRUE f65_in(x3) -> f67_in(x3) :|: TRUE f63_in(x8) -> f65_in(x8) :|: TRUE f198_in(x10) -> f63_in(x10) :|: TRUE ---------------------------------------- (123) Obligation: Rules: f67_in(.(x, x1)) -> f198_in(x1) :|: TRUE f65_in(x3) -> f67_in(x3) :|: TRUE f63_in(x8) -> f65_in(x8) :|: TRUE f198_in(x10) -> f63_in(x10) :|: TRUE ---------------------------------------- (124) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (125) Obligation: Rules: f63_in(.(x:0, x1:0)) -> f63_in(x1:0) :|: TRUE ---------------------------------------- (126) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (127) Obligation: Rules: f63_in(.(x:0, x1:0)) -> f63_in(x1:0) :|: TRUE ---------------------------------------- (128) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f63_in(.(x:0, x1:0)) -> f63_in(x1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (129) Obligation: Termination digraph: Nodes: (1) f63_in(.(x:0, x1:0)) -> f63_in(x1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (130) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: .(x1, x2) -> .(x2) ---------------------------------------- (131) Obligation: Rules: f63_in(.(x1:0)) -> f63_in(x1:0) :|: TRUE ---------------------------------------- (132) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f63_in(VARIABLE) .(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (133) Obligation: Rules: f63_in(.(x1:0)) -> f63_in(x1:0) ---------------------------------------- (134) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: f63_in(.(x1:0)) -> f63_in(x1:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (136) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *f63_in(.(x1:0)) -> f63_in(x1:0) The graph contains the following edges 1 > 1 ---------------------------------------- (137) YES ---------------------------------------- (138) Obligation: Rules: f368_out(T47) -> f64_out(T47) :|: TRUE f64_in(x) -> f368_in(x) :|: TRUE f368_in(x1) -> f370_in(x1) :|: TRUE f369_out(x2) -> f368_out(x2) :|: TRUE f370_out(x3) -> f368_out(x3) :|: TRUE f368_in(x4) -> f369_in(x4) :|: TRUE f65_in(T40) -> f67_in(T40) :|: TRUE f67_out(x5) -> f65_out(x5) :|: TRUE f65_in(x6) -> f66_in(x6) :|: TRUE f66_out(x7) -> f65_out(x7) :|: TRUE f63_out(T71) -> f198_out(T71) :|: TRUE f198_in(x8) -> f63_in(x8) :|: TRUE f81_in -> f81_out :|: TRUE f433_in(T94) -> f434_in(T94) :|: TRUE f433_in(x9) -> f435_in(x9) :|: TRUE f434_out(x10) -> f433_out(x10) :|: TRUE f435_out(x11) -> f433_out(x11) :|: TRUE f64_out(x12) -> f432_out(x12) :|: TRUE f432_in(x13) -> f64_in(x13) :|: TRUE f374_in(x14) -> f431_in(x14) :|: TRUE f431_out(x15) -> f432_in(x15) :|: TRUE f432_out(x16) -> f374_out(x16) :|: TRUE f200_out -> f67_out(x17) :|: TRUE f198_out(x18) -> f67_out(.(x19, x18)) :|: TRUE f67_in(.(x20, x21)) -> f198_in(x21) :|: TRUE f67_in(x22) -> f200_in :|: TRUE f436_in -> f436_out :|: TRUE f65_out(x23) -> f63_out(x23) :|: TRUE f63_in(x24) -> f65_in(x24) :|: TRUE f439_out(T124) -> f435_out(.(T123, T124)) :|: TRUE f435_in(.(x25, x26)) -> f439_in(x26) :|: TRUE f440_out -> f435_out(x27) :|: TRUE f435_in(x28) -> f440_in :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x29) -> f4_in(x29) :|: TRUE f441_in -> f441_out :|: TRUE f442_out -> f370_out(x30) :|: TRUE f370_in(x31) -> f442_in :|: TRUE f441_out -> f370_out(T135) :|: TRUE f370_in(x32) -> f441_in :|: TRUE f61_in(x33) -> f63_in(x33) :|: TRUE f63_out(x34) -> f64_in(x35) :|: TRUE f64_out(x36) -> f61_out(x37) :|: TRUE f369_in(x38) -> f375_in :|: TRUE f369_in(x39) -> f374_in(x39) :|: TRUE f375_out -> f369_out(x40) :|: TRUE f374_out(x41) -> f369_out(x41) :|: TRUE f434_in(x42) -> f437_in :|: TRUE f434_in(.(T114, T115)) -> f436_in :|: TRUE f437_out -> f434_out(x43) :|: TRUE f436_out -> f434_out(.(x44, x45)) :|: TRUE f439_in(x46) -> f431_in(x46) :|: TRUE f431_out(x47) -> f439_out(x47) :|: TRUE f42_out(T17) -> f40_out(T17) :|: TRUE f40_in(x48) -> f42_in(x48) :|: TRUE f34_out(x49) -> f4_out(x49) :|: TRUE f35_out(x50) -> f4_out(x50) :|: TRUE f4_in(x51) -> f35_in(x51) :|: TRUE f4_in(x52) -> f34_in(x52) :|: TRUE f41_in(x53) -> f3_in(x53) :|: TRUE f3_out(x54) -> f41_out(x54) :|: TRUE f41_out(x55) -> f36_out(x55) :|: TRUE f40_out(x56) -> f41_in(x56) :|: TRUE f36_in(x57) -> f40_in(x57) :|: TRUE f431_in(x58) -> f433_in(x58) :|: TRUE f433_out(x59) -> f431_out(x59) :|: TRUE f42_in(x60) -> f61_in(x60) :|: TRUE f62_out -> f42_out(x61) :|: TRUE f42_in(x62) -> f62_in :|: TRUE f61_out(x63) -> f42_out(x63) :|: TRUE f81_out -> f66_out(.(T61, T62)) :|: TRUE f90_out -> f66_out(x64) :|: TRUE f66_in(.(x65, x66)) -> f81_in :|: TRUE f66_in(x67) -> f90_in :|: TRUE f34_in(x68) -> f37_in :|: TRUE f34_in(x69) -> f36_in(x69) :|: TRUE f37_out -> f34_out(x70) :|: TRUE f36_out(x71) -> f34_out(x71) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (139) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f368_out(T47) -> f64_out(T47) :|: TRUE f64_in(x) -> f368_in(x) :|: TRUE f368_in(x1) -> f370_in(x1) :|: TRUE f369_out(x2) -> f368_out(x2) :|: TRUE f370_out(x3) -> f368_out(x3) :|: TRUE f368_in(x4) -> f369_in(x4) :|: TRUE f65_in(T40) -> f67_in(T40) :|: TRUE f67_out(x5) -> f65_out(x5) :|: TRUE f65_in(x6) -> f66_in(x6) :|: TRUE f66_out(x7) -> f65_out(x7) :|: TRUE f63_out(T71) -> f198_out(T71) :|: TRUE f198_in(x8) -> f63_in(x8) :|: TRUE f81_in -> f81_out :|: TRUE f433_in(T94) -> f434_in(T94) :|: TRUE f433_in(x9) -> f435_in(x9) :|: TRUE f434_out(x10) -> f433_out(x10) :|: TRUE f435_out(x11) -> f433_out(x11) :|: TRUE f64_out(x12) -> f432_out(x12) :|: TRUE f432_in(x13) -> f64_in(x13) :|: TRUE f374_in(x14) -> f431_in(x14) :|: TRUE f431_out(x15) -> f432_in(x15) :|: TRUE f432_out(x16) -> f374_out(x16) :|: TRUE f198_out(x18) -> f67_out(.(x19, x18)) :|: TRUE f67_in(.(x20, x21)) -> f198_in(x21) :|: TRUE f436_in -> f436_out :|: TRUE f65_out(x23) -> f63_out(x23) :|: TRUE f63_in(x24) -> f65_in(x24) :|: TRUE f439_out(T124) -> f435_out(.(T123, T124)) :|: TRUE f435_in(.(x25, x26)) -> f439_in(x26) :|: TRUE f3_in(x29) -> f4_in(x29) :|: TRUE f441_in -> f441_out :|: TRUE f441_out -> f370_out(T135) :|: TRUE f370_in(x32) -> f441_in :|: TRUE f61_in(x33) -> f63_in(x33) :|: TRUE f63_out(x34) -> f64_in(x35) :|: TRUE f64_out(x36) -> f61_out(x37) :|: TRUE f369_in(x39) -> f374_in(x39) :|: TRUE f374_out(x41) -> f369_out(x41) :|: TRUE f434_in(.(T114, T115)) -> f436_in :|: TRUE f436_out -> f434_out(.(x44, x45)) :|: TRUE f439_in(x46) -> f431_in(x46) :|: TRUE f431_out(x47) -> f439_out(x47) :|: TRUE f42_out(T17) -> f40_out(T17) :|: TRUE f40_in(x48) -> f42_in(x48) :|: TRUE f4_in(x52) -> f34_in(x52) :|: TRUE f41_in(x53) -> f3_in(x53) :|: TRUE f40_out(x56) -> f41_in(x56) :|: TRUE f36_in(x57) -> f40_in(x57) :|: TRUE f431_in(x58) -> f433_in(x58) :|: TRUE f433_out(x59) -> f431_out(x59) :|: TRUE f42_in(x60) -> f61_in(x60) :|: TRUE f61_out(x63) -> f42_out(x63) :|: TRUE f81_out -> f66_out(.(T61, T62)) :|: TRUE f66_in(.(x65, x66)) -> f81_in :|: TRUE f34_in(x69) -> f36_in(x69) :|: TRUE ---------------------------------------- (140) Obligation: Rules: f368_out(T47) -> f64_out(T47) :|: TRUE f64_in(x) -> f368_in(x) :|: TRUE f368_in(x1) -> f370_in(x1) :|: TRUE f369_out(x2) -> f368_out(x2) :|: TRUE f370_out(x3) -> f368_out(x3) :|: TRUE f368_in(x4) -> f369_in(x4) :|: TRUE f65_in(T40) -> f67_in(T40) :|: TRUE f67_out(x5) -> f65_out(x5) :|: TRUE f65_in(x6) -> f66_in(x6) :|: TRUE f66_out(x7) -> f65_out(x7) :|: TRUE f63_out(T71) -> f198_out(T71) :|: TRUE f198_in(x8) -> f63_in(x8) :|: TRUE f81_in -> f81_out :|: TRUE f433_in(T94) -> f434_in(T94) :|: TRUE f433_in(x9) -> f435_in(x9) :|: TRUE f434_out(x10) -> f433_out(x10) :|: TRUE f435_out(x11) -> f433_out(x11) :|: TRUE f64_out(x12) -> f432_out(x12) :|: TRUE f432_in(x13) -> f64_in(x13) :|: TRUE f374_in(x14) -> f431_in(x14) :|: TRUE f431_out(x15) -> f432_in(x15) :|: TRUE f432_out(x16) -> f374_out(x16) :|: TRUE f198_out(x18) -> f67_out(.(x19, x18)) :|: TRUE f67_in(.(x20, x21)) -> f198_in(x21) :|: TRUE f436_in -> f436_out :|: TRUE f65_out(x23) -> f63_out(x23) :|: TRUE f63_in(x24) -> f65_in(x24) :|: TRUE f439_out(T124) -> f435_out(.(T123, T124)) :|: TRUE f435_in(.(x25, x26)) -> f439_in(x26) :|: TRUE f3_in(x29) -> f4_in(x29) :|: TRUE f441_in -> f441_out :|: TRUE f441_out -> f370_out(T135) :|: TRUE f370_in(x32) -> f441_in :|: TRUE f61_in(x33) -> f63_in(x33) :|: TRUE f63_out(x34) -> f64_in(x35) :|: TRUE f64_out(x36) -> f61_out(x37) :|: TRUE f369_in(x39) -> f374_in(x39) :|: TRUE f374_out(x41) -> f369_out(x41) :|: TRUE f434_in(.(T114, T115)) -> f436_in :|: TRUE f436_out -> f434_out(.(x44, x45)) :|: TRUE f439_in(x46) -> f431_in(x46) :|: TRUE f431_out(x47) -> f439_out(x47) :|: TRUE f42_out(T17) -> f40_out(T17) :|: TRUE f40_in(x48) -> f42_in(x48) :|: TRUE f4_in(x52) -> f34_in(x52) :|: TRUE f41_in(x53) -> f3_in(x53) :|: TRUE f40_out(x56) -> f41_in(x56) :|: TRUE f36_in(x57) -> f40_in(x57) :|: TRUE f431_in(x58) -> f433_in(x58) :|: TRUE f433_out(x59) -> f431_out(x59) :|: TRUE f42_in(x60) -> f61_in(x60) :|: TRUE f61_out(x63) -> f42_out(x63) :|: TRUE f81_out -> f66_out(.(T61, T62)) :|: TRUE f66_in(.(x65, x66)) -> f81_in :|: TRUE f34_in(x69) -> f36_in(x69) :|: TRUE ---------------------------------------- (141) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (142) Obligation: Rules: f65_in(.(x20:0, x21:0)) -> f65_in(x21:0) :|: TRUE f63_out(x34:0) -> f64_in(x35:0) :|: TRUE f431_out(x47:0) -> f431_out(.(T123:0, x47:0)) :|: TRUE f431_out(x15:0) -> f64_in(x15:0) :|: TRUE f63_out(T71:0) -> f63_out(.(x19:0, T71:0)) :|: TRUE f433_in(.(x25:0, x26:0)) -> f433_in(x26:0) :|: TRUE f433_in(.(T114:0, T115:0)) -> f431_out(.(x44:0, x45:0)) :|: TRUE f64_in(x:0) -> f368_out(T135:0) :|: TRUE f368_out(T47:0) -> f368_out(T47:0) :|: TRUE f65_in(.(x65:0, x66:0)) -> f63_out(.(T61:0, T62:0)) :|: TRUE f368_out(x) -> f65_in(x1) :|: TRUE f64_in(x2) -> f433_in(x2) :|: TRUE ---------------------------------------- (143) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (144) Obligation: Rules: f65_in(.(x20:0, x21:0)) -> f65_in(x21:0) :|: TRUE f63_out(x34:0) -> f64_in(x35:0) :|: TRUE f431_out(x47:0) -> f431_out(.(T123:0, x47:0)) :|: TRUE f431_out(x15:0) -> f64_in(x15:0) :|: TRUE f63_out(T71:0) -> f63_out(.(x19:0, T71:0)) :|: TRUE f433_in(.(x25:0, x26:0)) -> f433_in(x26:0) :|: TRUE f433_in(.(T114:0, T115:0)) -> f431_out(.(x44:0, x45:0)) :|: TRUE f64_in(x:0) -> f368_out(T135:0) :|: TRUE f368_out(T47:0) -> f368_out(T47:0) :|: TRUE f65_in(.(x65:0, x66:0)) -> f63_out(.(T61:0, T62:0)) :|: TRUE f368_out(x) -> f65_in(x1) :|: TRUE f64_in(x2) -> f433_in(x2) :|: TRUE ---------------------------------------- (145) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f65_in(.(x20:0, x21:0)) -> f65_in(x21:0) :|: TRUE (2) f63_out(x34:0) -> f64_in(x35:0) :|: TRUE (3) f431_out(x47:0) -> f431_out(.(T123:0, x47:0)) :|: TRUE (4) f431_out(x15:0) -> f64_in(x15:0) :|: TRUE (5) f63_out(T71:0) -> f63_out(.(x19:0, T71:0)) :|: TRUE (6) f433_in(.(x25:0, x26:0)) -> f433_in(x26:0) :|: TRUE (7) f433_in(.(T114:0, T115:0)) -> f431_out(.(x44:0, x45:0)) :|: TRUE (8) f64_in(x:0) -> f368_out(T135:0) :|: TRUE (9) f368_out(T47:0) -> f368_out(T47:0) :|: TRUE (10) f65_in(.(x65:0, x66:0)) -> f63_out(.(T61:0, T62:0)) :|: TRUE (11) f368_out(x) -> f65_in(x1) :|: TRUE (12) f64_in(x2) -> f433_in(x2) :|: TRUE Arcs: (1) -> (1), (10) (2) -> (8), (12) (3) -> (3), (4) (4) -> (8), (12) (5) -> (2), (5) (6) -> (6), (7) (7) -> (3), (4) (8) -> (9), (11) (9) -> (9), (11) (10) -> (2), (5) (11) -> (1), (10) (12) -> (6), (7) This digraph is fully evaluated! ---------------------------------------- (146) Obligation: Termination digraph: Nodes: (1) f65_in(.(x20:0, x21:0)) -> f65_in(x21:0) :|: TRUE (2) f368_out(x) -> f65_in(x1) :|: TRUE (3) f368_out(T47:0) -> f368_out(T47:0) :|: TRUE (4) f64_in(x:0) -> f368_out(T135:0) :|: TRUE (5) f431_out(x15:0) -> f64_in(x15:0) :|: TRUE (6) f431_out(x47:0) -> f431_out(.(T123:0, x47:0)) :|: TRUE (7) f433_in(.(T114:0, T115:0)) -> f431_out(.(x44:0, x45:0)) :|: TRUE (8) f433_in(.(x25:0, x26:0)) -> f433_in(x26:0) :|: TRUE (9) f64_in(x2) -> f433_in(x2) :|: TRUE (10) f63_out(x34:0) -> f64_in(x35:0) :|: TRUE (11) f63_out(T71:0) -> f63_out(.(x19:0, T71:0)) :|: TRUE (12) f65_in(.(x65:0, x66:0)) -> f63_out(.(T61:0, T62:0)) :|: TRUE Arcs: (1) -> (1), (12) (2) -> (1), (12) (3) -> (2), (3) (4) -> (2), (3) (5) -> (4), (9) (6) -> (5), (6) (7) -> (5), (6) (8) -> (7), (8) (9) -> (7), (8) (10) -> (4), (9) (11) -> (10), (11) (12) -> (10), (11) This digraph is fully evaluated! ---------------------------------------- (147) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: .(x1, x2) -> .(x2) ---------------------------------------- (148) Obligation: Rules: f65_in(.(x21:0)) -> f65_in(x21:0) :|: TRUE f368_out(x) -> f65_in(x1) :|: TRUE f368_out(T47:0) -> f368_out(T47:0) :|: TRUE f64_in(x:0) -> f368_out(T135:0) :|: TRUE f431_out(x15:0) -> f64_in(x15:0) :|: TRUE f431_out(x47:0) -> f431_out(.(x47:0)) :|: TRUE f433_in(.(T115:0)) -> f431_out(.(x45:0)) :|: TRUE f433_in(.(x26:0)) -> f433_in(x26:0) :|: TRUE f64_in(x2) -> f433_in(x2) :|: TRUE f63_out(x34:0) -> f64_in(x35:0) :|: TRUE f63_out(T71:0) -> f63_out(.(T71:0)) :|: TRUE f65_in(.(x66:0)) -> f63_out(.(T62:0)) :|: TRUE